Users Guide to the

R-to-MOSEK Optimization Interface

Made by:
MOSEK ApS <support@mosek.com>
Package Rmosek v7.0.4
February 14, 2014
MOSEK ApS
Scope
The MOSEK optimization library provides a solver with industrial strength capable of
solving huge convex problems: linear, quadratic, conic, continuous and mixed integer. The
development of the Rmosek package was initiated to give access to MOSEK through the
R-language. The interface is simple to learn and utilize despite its many features, and
thus provides a perfect tool for research and educational projects. However, it should be
noted that it does not reect the full capabilities of the MOSEK optimization library. The
advanced user is welcome to customize or added features to this open-source interface.
2
Contents
1 Installation 5
1.1 Windows platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 UNIX-alike platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 A guided tour 14
2.1 Help: How do I use the interface? . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Solving LP problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 Switching Optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Hot-starting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.5 Complete Best Practice example . . . . . . . . . . . . . . . . . . . . 29
2.3 Conic Quadratic Programming . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.1 Solving CQP problems . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.3 Notes on Quadratic Programming (QP/QCQP) . . . . . . . . . . . . 40
2.4 Semidenite Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4.1 Solving SDP problems . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Mixed Integer Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5.1 Solving MILP and MICQP problems . . . . . . . . . . . . . . . . . . 44
2.5.2 Hot-starting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5.3 Termination criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6 Parameter settings in MOSEK . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.7 Exporting and Importing optimization models . . . . . . . . . . . . . . . . . 51
2.7.1 The write and read functionality . . . . . . . . . . . . . . . . . . . . 51
2.7.2 Writing immediately before/after optimizing . . . . . . . . . . . . . 54
2.8 Frequently Asked Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.8.1 How to make Rmosek silent? . . . . . . . . . . . . . . . . . . . . . . 56
2.8.2 How to force a premature termination? . . . . . . . . . . . . . . . . 56
2.8.3 How to change the termination criteria? . . . . . . . . . . . . . . . . 56
2.8.4 How to nd the objective value and other information items? . . . . 57
2.8.5 How to report a bug or get further help? . . . . . . . . . . . . . . . . 57
3 Advanced topics 58
3.1 Parallel optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.1 The multicore package . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Quadratic Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.1 Solving QP problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3
3.3 Separable Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.1 Solving SCOPT problems . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.2 Convexity, dierentiability and safe bounds . . . . . . . . . . . . . . 66
3.3.3 Importing and exporting modelles . . . . . . . . . . . . . . . . . . . 67
4 Developers corner 68
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Limitations and Missing Features . . . . . . . . . . . . . . . . . . . . . . . . 69
A Command reference 70
B Conic transformations 77
B.1 The Quadratic Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4
INSTALLATION
1 Installation
1.1 Windows platforms
The interface documented here is part of the Rmosek package distributed from http:
//rmosek.r-forge.r-project.org/. Notice, however, that a pre-compiled binary version
of it has not been distributed
1
. Thus a small amount of conguration is necessary in order
to install the package. It is not very dicult, but besides the basic installation of R you
will need two pieces of software readily available through the Internet. In summary:
MOSEK (the optimization library we interface to)
2
Rtools (the tools needed for R package development)
The following is a step-by-step guide through the installation of the package. We skip
the installation of MOSEK as it has been well documented elsewhere and assume that
the target machine already has a working installation. Please refer to the documentation
available at http://mosek.com for the verication of this installation. The basic test for
this is:
1. Does calling mosek -f from the Windows CMD cause errors?
For additional help installing this interface, the section on packages in the R for
Windows FAQ available on the CRAN website may be useful. The R Installation and
Administration manual, also published by CRAN on their website, is another good source
of information.
1
Such a binary would have to be built individually for each version of R and MOSEK.
2
Check http://mosek.com on free licenses for academic use and trial evaluation periods. For commercial
use, licenses needed to handle a problem with MOSEK is also those needed when using Rmosek.
5
1.1 Windows platforms INSTALLATION
Setting up the target machine
Assuming a working installation of R and MOSEK on the machine targeted for the R-
to-MOSEK interface, the rst step is to download Rtools for Windows
3
. From this
program you will need to install the component called R toolset, the Cygwin DLLs, and
the R toolchain.
R toolset and Cygwin DLLs wil l extend the Windows CMD with Unix-style commands,
while the R toolchain (based on MinGW compilers) makes it possible to compile the
C++ source code in the package. These components can be replaced by any other Unix-
style shel l and C++ build chain, but in the remaining guide the use of Rtools wil l be
assumed.
After the installation of Rtools you will have to set the Windows environment variable
called PATH, in order to utilize the components. Assuming that the home directory of
the Rtools installation was c:\Rtools, the entries shown below will have to be added to
the existing PATH variable. Note that all entries in the PATH variable must be separated
by a semicolon (;), and that all these entries have to represent folders that exist on the
target machine.
Add c:\Rtools\bin; to enable the R toolset and Cygwin DLLs.
Add c:\Rtools\gcc-VERSION\bin; (for some VERSION) to enable the R toolchain.
That was it, but before we move on to the installation process, please ensure that the
PATH variable also contains the bin folder of all MOSEK installations (32 and/or 64
bit) that you wish this interface to target. This is necessary for automatic conguration
to work, and could look something like:
C:\Progra1\Mosek\7\tools\platform\win64x86\bin
Installing the package with automatic conguration
Automatic conguration works equivalently to calling where mosek in the Windows CMD.
It searches the environment variable called PATH for a folder with an executable called
mosek. Note that if more than one such folder exists, only the one mentioned rst in the
PATH variable is chosen. It then determines the most ordinary of the available optimization
libraries within this folder (typically mosek.lib or mosek64.lib), along with other relevant
information. This conguration should work for all users installing the package on a single
architecture (64 or 32 bit) and only requiring the ordinary optimization library. Otherwise,
manual conguration of the package will be needed.
3
Newest release of Rtools available at http://cran.r-project.org/bin/windows/Rtools
6
1.1 Windows platforms INSTALLATION
Now open R for the architecture (64 or 32 bit) you wish to install the package on. Make sure
all your packages are up to date by writing update.packages(), and execute a command
similar to the one shown below. This will install the Rmosek package:
install.packages("Rmosek", type="source", INSTALL_opts="--no-multiarch",
repos="http://download.mosek.com/R/7")
Note that this package will have to be installed from source as it needs a static link to
the MOSEK optimization library. This unfortunately means that dependencies (i.e. the
Matrix package) will also be installed from source if new releases are available. Since it
is more time-consuming to install the Matrix package from source, it is recommended to
start with a call to update.packages(). The availability of the Matrix package should
not be a problem, as it has been part of the standard R installation since version 2.9.0.
Installing the package with manual conguration
If the automatic conguration does not suit your particular needs, or fails for some reason,
a manual conguration may work instead. Unfortunately the configure.vars parameter
of the install.packages command does not work on Windows, meaning that the les of
the Rmosek source package will have to be edited. This can be dicult for non-savvy
users, but have hopefully been documented suciently here.
How to manual ly congure the Rmosek package?
When you download the package from http://rmosek.r-forge.r-project.org/, it
comes in a compressed archive called Rmosek_VERSION.tar.gz for some version number
(VERSION). In order to congure the package, you will have to go through the following
steps. A more thorough explanation of each step will be given afterwards.
1. Extract the archive into a directory.
2. Setup the local system descriptors, Localsys.txt, for each of the sub-architectures you
will be using: 64 bit (e.g. x64) and/or 32 bit (e.g. i386). These les come with a
guide written into them that clearly states how this should be done.
3. Compress the directory back into an archive.
The rst step is to extract the Rmosek_VERSION.tar.gz archive that you downloaded.
This can either be done by using one of the many tools freely available online
4
, or the
4
Search the web for the keywords: Windows, Packing, Unpacking, Tar, Gz
7
1.1 Windows platforms INSTALLATION
tar-command that was installed with Rtools. If you choose to use the tar-command, you
can extract the package by opening Windows CMD and executing a command similar to:
tar --no-same-owner -zxvf LOCATION\Rmosek_VERSION.tar.gz
Remember to exchange LOCATION and VERSION with the correct values. For those
not familiar with Windows CMD, we recommend the use of an external tool as this
conguration can then be performed entirely within Windows Explorer.
The second step is to tell the package where to nd the MOSEK optimization library. Open
the extracted directory called Rmosek in either Windows CMD or Windows Explorer, and
navigate to the subdirectory called src\setup. If you want to install a 64 bit version of the
package (making an interface between 64 bit R and 64 bit MOSEK), open the folder x64
and follow the guide in Localsys.txt. If you want to install a 32 bit version of the package
(making an interface between 32 bit R and 32 bit MOSEK), open instead the folder i386
and follow the guide in this Localsys.txt. An example of this is shown in Figure 1.
The third and nal step is to compress the (previously extracted and now altered) directory
called Rmosek, back into the Rmosek_VERSION.tar.gz archive. Again you can either make
use of the external tools from step one, or open the Windows CMD and execute a command
similar to:
tar -zcvf Rmosek_VERSION.tar.gz Rmosek
How to instal l the manual ly congured Rmosek package?
Open R, either the console or the graphical user interface, for the architecture (64 or 32
bit) you wish to install the package on. Make sure all your packages are up to date by
writing update.packages(), and execute the following command to install the Rmosek
package:
install.packages("LOCATION\Rmosek_VERSION.tar.gz", repos=NULL,
type="source", INSTALL_opts="--no-multiarch")
Remember to exchange LOCATION and VERSION with the correct values. The rst
argument should be the path to your manually congured package. The second argument
tells that the package is local and not in an online repository. The third argument tells
that it is a source package and so should be compiled. The fourth and nal argument
species that you only wish to install the package to the specic architecture (64 or 32 bit)
of the opened R program. If you wish to install on both architectures, and did congure
the package for both 32 and 64 bit, simply remove this last argument.
Notice that if you wish to uninstall the Rmosek package at some point, this can be done
as for any other package with the command remove.packages("Rmosek").
8
1.1 Windows platforms INSTALLATION
Figure 1: The contents of le ./src/setup/x64/Localsys.txt
##
## Greetings user of the R-to -MOSEK interface!
##
## If you are sitting on a WINDOWS 64bit platform , this is the file that
## you will have to setup before this package can be installed.
## (see e.g. the R-to -MOSEK userguide)
##
##################
## Step 1 of 2 ##
##################
## Please substitute [MOSEK_HOME_PATH] below , with the path to the
## platform -specific folder within the MOSEK installation you want to
## use. Note that this path should contain a "bin" and a "h" folder.
## --------------------------------------------------------------------
## For example you can write:
## PKG_MOSEKHOME=C:\ Progra ~1\ Mosek \7\ tools\platform\win64x86
##
## If your computer contains the two directories:
## C:\ Progra ~1\ Mosek \7\ tools\platform\win64x86\bin
## C:\ Progra ~1\ Mosek \7\ tools\platform\win64x86\h
## --------------------------------------------------------------------
PKG_MOSEKHOME =[ MOSEK_HOME_PATH]
##################
## Step 2 of 2 ##
##################
## Please substitute [MOSEK_LIB_FILE] below , with the name of the library
## you wish to use within the "bin" folder of your PKG_MOSEKHOME path.
## This "bin" folder must contain a file called [MOSEK_LIB_FILE ].lib.
## --------------------------------------------------------------------
## Continuing the example from above , you can write:
## PKG_MOSEKLIB=mosek64_7_0
##
## If your computer contains the file:
## C:\ Progra ~1\ Mosek \7\ tools\platform\win64x86\bin\mosek64_7_0.lib
## --------------------------------------------------------------------
PKG_MOSEKLIB =[ MOSEK_LIB_FILE]
9
1.2 UNIX-alike platforms INSTALLATION
1.2 UNIX-alike platforms
The interface documented here is part of the Rmosek package distributed from http:
//rmosek.r-forge.r-project.org/. Notice, however, that a pre-compiled binary version
of it has not been distributed
5
. Thus a small amount of conguration is necessary in order
to install the package. It is not very dicult, but besides the basic installation of R you
will need MOSEK (the optimization library we interface to)
6
readily available through
the Internet.
The following is a step-by-step guide through the installation of the package. We skip
the installation of MOSEK as it has been well documented elsewhere and assume that
the target machine already has a working installation. Please refer to the documentation
available at http://mosek.com for the verication of this installation. The basic test is:
1. Does calling mosek from a terminal window cause errors?
For additional help on installing this interface, the section on installing packages in the
manual R Installation and Administration, published on the CRAN website, is a good
source of information.
5
Such a binary would have to be built individually for each version of R and MOSEK.
6
Check http://mosek.com on free licenses for academic use and trial evaluation periods. For commercial
use, licenses needed to handle a problem with MOSEK is also those needed when using Rmosek.
10
1.2 UNIX-alike platforms INSTALLATION
Setting up the target machine
We assume here that you have a working installation of R and MOSEK on the machine
targeted for the R-to-MOSEK interface. The architectures (32 or 64 bit) of these two
programs must be exactly the same for consistency. We further assume that the target
machine have a build chain (e.g. GCC compilers for C/C++) installed.
For automatic conguration to work, the PATH variable should contain the bin folder of
the single MOSEK installation (32 or 64 bit) that you wish this interface to target. This
could look something like:
/mosek/7/tools/platform/linux64x86/bin
If more than one bin folder from a MOSEK installation are specied, only the rst one
will be found by automatic conguration. Additional architectures can be added afterwards
if necessary.
Installing the package with automatic conguration
Automatic conguration works equivalently to calling which mosek in a terminal window.
It searches the environment variable called PATH for a folder with an executable called
mosek. Note that if more than one such folder exists, only the one mentioned rst in the
PATH variable is chosen. It then determines the most ordinary of the available optimization
libraries within this folder (typically libmosek or libmosek64 with the extension .so or
.dylib), along with other relevant information. This conguration should work for all users
only installing the package on a single architecture (64 or 32 bit) and only requiring the
ordinary optimization library. Otherwise, manual conguration of the package will be
needed.
Now open R for the architecture (64 or 32 bit) you wish to install the package on. Make sure
all your packages are up to date by writing update.packages(), and execute a command
similar to the one shown below. This will install the Rmosek package:
install.packages("Rmosek", type="source", INSTALL_opts="--no-multiarch",
repos="http://download.mosek.com/R/7")
Note that this package will have to be installed from source as it needs a static link to
the MOSEK optimization library. This unfortunately means that dependencies (i.e. the
Matrix package) will also be installed from source if new releases are available. Since it
is more time-consuming to install the Matrix package from source, it is recommended to
start with a call to update.packages(). The availability of the Matrix package should
not be a problem, as it has been part of the standard R installation since version 2.9.0.
11
1.2 UNIX-alike platforms INSTALLATION
Installing the package with manual conguration
Open R for one of the architectures (64 or 32 bit) you wish to install the package
on. Make sure all your packages are up to date by writing update.packages(),
and execute a command similar to the one shown below, with a correct denition of
PKG_MOSEKHOME and PKG_MOSEKLIB. This will install the Rmosek package:
install.packages("Rmosek", type="source", INSTALL_opts="--no-multiarch",
repos="http://download.mosek.com/R/7",
configure.vars="PKG_MOSEKHOME=... PKG_MOSEKLIB=...")
Remember to exchange the ... of both PKG_MOSEKHOME and PKG_MOSEKLIB
with the correct values, explained as follows. The denition of the rst argument,
PKG_MOSEKHOME, should be the folder in your MOSEK installation, containing a
bin and h subdirectory for the platform and architecture matching that of the opened
R program. This could for instance look something like:
/mosek/7/tools/platform/linux64x86
Notice that if the folder denition contains spaces you will either have to wrap the denition
in single-quotes (preventing auto-expansions such as the above), or add backslashes in
front of all spaces.
The denition of the argument PKG_MOSEKLIB should be the name of the
optimization library in the bin subdirectory that you wish to utilize in the Rmosek
package. This library will be statically linked to the package after a successful installation.
Note that the name of the optimization library should be specied without the lib prex,
and without its le-extension. The PKG_MOSEKLIB would thus normally be either
mosek or mosek64 (linking to respectively libmosek.so and libmosek64.so, or respectively
libmosek.dylib and libmosek64.dylib, depending on the Unix-alike system). Using mosek64
requires a 64 bit version of the opened R program and MOSEK installation, while mosek
implies 32 bit.
How to instal l on multiple architectures?
If you wish to install the Rmosek package on multiple sub-architectures, you will rst have
to follow the above guide and install the package on one of the architectures (e.g. 32 or 64
bit). Afterwards as explained here, it can then be extended to other sub-architectures. This
is also explained in R Installation and Administration (published on the CRAN website),
in the subsection Multiple sub-architectures under Installing packages. For this to work
you will need a preinstalled version of R and MOSEK in all the sub-architectures you wish
this package to work with.
12
1.2 UNIX-alike platforms INSTALLATION
Open R in the sub-architecture you wish to extend your installation to, and execute a
command similar to the one shown below:
install.packages("Rmosek", type="source", libs_only=TRUE,
repos="http://download.mosek.com/R/7",
configure.vars="PKG_MOSEKHOME=... PKG_MOSEKLIB=...")
Remember to exchange ... of the variables PKG_MOSEKHOME and PKG_MOSEKLIB,
so that declared MOSEK installation always correspond to the current sub-architecture of
the opened R program.
How to instal l from an oine location?
This is almost the same a described above except that you would have to add the argument
repos=NULL to tell R that it should not use an online repository. Also, instead of writing
the package name "Rmosek", you should write the entire le-location of the package source
le which should look something like "LOCATION/Rmosek_VERSION.tar.gz".
What are the command-line equivalents?
Sometimes you do not want to open R for all sub-architectures, but instead perform the
installation directly from the console. Notice though, that this will require you to download
the package source as you will only be able to install from an oine location in this way. In
the two commands below you will have to replace SUBARCH_PATH, DIR and VERSION
with the correct values, but doing so should be straight forward. Notice that on newer
versions of R (>= 2.12), you can also use the call R --arch SUBARCH, instead of specifying
the SUBARCH_PATH. Remember to exchange ... of the variables PKG_MOSEKHOME
and PKG_MOSEKLIB, with the correct values as previously explained.
For the rst architecture:
SUBARCH_PATH/R CMD INSTALL DIR/Rmosek_VERSION.tar.gz --no-multiarch
--configure-vars="PKG_MOSEKHOME=... PKG_MOSEKLIB=..."
For subsequent architectures:
SUBARCH_PATH/R CMD INSTALL DIR/Rmosek_VERSION.tar.gz --libs-only
--configure-vars="PKG_MOSEKHOME=... PKG_MOSEKLIB=..."
13
A GUIDED TOUR
2 A guided tour
2.1 Help: How do I use the interface?
To access the functions of the R-to-MOSEK interface, the Rmosek package have to be
loaded rst. Opening the R-console and typing the command:
require(Rmosek)
will load the package if it has been installed properly. If this does not work, you should
go back to Section 1 and check your installation. Note that the Matrix package will
automatically be loaded also as the Rmosek package depends on it for the representation
of sparse matrices. The Matrix package have been part of the standard R installation since
version 2.9.0, and should thus be available without further notice.
If the package loaded successfully without errors, you should now be able to use the
functions of the R-to-MOSEK interface. For example, the following function will return
the version of the MOSEK optimization library that this package is currently linked to:
mosek_version()
To see a summary of the Rmosek package with a complete list of all exported functions,
the following command can be used:
library(help="Rmosek")
In addition to mosek_version, this list also includes mosek, which is the main function
of the package able to solve a broad range of convex optimization problems using one
of the optimizers from MOSEK. When mosek is called for the rst time, a MOSEK
environment is acquired occupying one user license. Given the extra overhead of acquiring
and releasing licenses, this environment is typically held until R is terminated. If the
user wish to release the environment and occupied license earlier than this, a call to
mosek_clean will free such resources. To support other modeling languages, the functions
mosek_read and mosek_write is able to handle model les in several formats. The
functions mosek_lptoprob and mosek_qptoprob oers an alternative way of formulating
linear and quadratic programs.
NB! If you share a limited number of licenses among multiple users,
remember to call mosek_clean() - or terminate R - to release yours.
14
2.1 Help: How do I use the interface? A GUIDED TOUR
Being aware of how the user licenses of MOSEK are handled in this interface, we can now
proceed with guidance on how to use the available functions. For instance, if you wish
to know more about the mosek function, you would simple have to execute the following
command:
?mosek
This will give you a detailed description of the function, including a list of recognized
input arguments, an outline of the expected output, and an example of its usage. This
information will also be shown if you fail to call a function with the correct number of
input arguments. As an example, the input arguments and return value printed as part of
the information given when calling ?mosek, is shown in Figure 2.
Many of the input and output arguments are probably not relevant for your project and so
will not all be explained here, but instead elaborated in the sections to which they belong.
If you are new to the interface, you can simply to read on. If you on the other hand are
looking for something specically, it may be benecial to look at the brief summary in
Appendix A for references to more information.
Figure 2: Function manual summary: mosek
Solve an optimization problem
Usage:
r <- mosek(problem , opts = list ())
Arguments:
problem LIST
.. $sense STRING
..$c NUMERIC VECTOR
..$c0 NUMERIC (OPTIONAL)
..$A SPARSE MATRIX
..$bc NUMERIC MATRIX (2 rows)
..$bx NUMERIC MATRIX (2 rows)
.. $cones LIST MATRIX (2 rows) (OPTIONAL)
.. $bardim NUMERIC VECTOR (OPTIONAL)
.. $barc LIST (OPTIONAL)
.. $barA LIST (OPTIONAL)
.. $intsub NUMERIC VECTOR (OPTIONAL)
.. $qobj LIST (OPTIONAL)
.. $scopt LIST (OPTIONAL)
.. $iparam/$dparam/$sparam LIST (OPTIONAL)
....$<MSK_PARAM > STRING / NUMERIC (OPTIONAL)
..$sol LIST (OPTIONAL)
.... $itr/$bas/$int LIST (OPTIONAL)
15
2.1 Help: How do I use the interface? A GUIDED TOUR
opts LIST (OPTIONAL)
.. $verbose NUMERIC (OPTIONAL)
.. $usesol BOOLEAN (OPTIONAL)
.. $useparam BOOLEAN (OPTIONAL)
.. $soldetail NUMERIC (OPTIONAL)
.. $getinfo BOOLEAN (OPTIONAL)
.. $writebefore STRING (filepath) (OPTIONAL)
.. $writeafter STRING (filepath) (OPTIONAL)
Value:
r LIST
.. $response LIST
.... $code NUMERIC
.... $msg STRING
..$sol LIST
.... $itr/$bas/$int LIST (SOLVER DEPENDENT)
...... $solsta STRING
...... $prosta STRING
...... $skc STRING VECTOR
...... $skx STRING VECTOR
...... $skn STRING VECTOR (NOT in $bas)
...... $xc NUMERIC VECTOR
...... $xx NUMERIC VECTOR
...... $barx NUMERIC VECTOR (NOT in $bas)
...... $slc NUMERIC VECTOR (NOT IN $int)
...... $suc NUMERIC VECTOR (NOT IN $int)
...... $slx NUMERIC VECTOR (NOT IN $int)
...... $sux NUMERIC VECTOR (NOT IN $int)
...... $snx NUMERIC VECTOR (NOT IN $int/$bas)
...... $bars NUMERIC VECTOR (NOT in $int/$bas)
...... $pobjval NUMERIC *
...... $dobjval NUMERIC *(NOT IN $int)
...... $pobjbound NUMERIC *($int ONLY)
...... $maxinfeas LIST *
........ $pbound NUMERIC *
........ $peq NUMERIC *
........ $pcone NUMERIC *(NOT in $bas)
........ $dbound NUMERIC *(NOT in $int)
........ $deq NUMERIC *(NOT in $int)
........ $dcone NUMERIC *(NOT in $int/$bas)
........ $int NUMERIC *($int ONLY)
.. $iinfo/$dinfo LIST *
....$<MSK_INFO > NUMERIC *
*Starred items must be requested using an option.
16
2.2 Linear Programming A GUIDED TOUR
2.2 Linear Programming
2.2.1 Solving LP problems
Linear optimization problems
7
posses a strong set of properties that makes them easy to
solve. Any linear optimization problem can be written as shown below, with variables
x R
n
, constraint matrix A R
mn
, objective coecients c R
n
, objective constant
c
0
R, lower and upper constraint bounds l
c
R
m
and u
c
R
m
, and lower and upper
variable bounds l
x
R
n
and u
x
R
n
. This is the so called primal problem.
minimize c
T
x + c
0
subject to l
c
Ax u
c
,
l
x
x u
x
.
(2.1)
All LP problems can be written on this form where e.g. equality constraint will be given the
same upper and lower bound, but what if you have an upper-bounded constraint with no
lower-bound? To exclude such bounds, the interface utilizes the R constant Inf and allows
you to specify lower bounds of minus innity (-Inf ) and upper bounds of plus innity (Inf ),
that will render bounds as non-existing. The interface will always expect LP problems on
this form, and will accordingly set the dual variables of the non-existing bounds to zero to
satisfy complementarity - a condition for optimality.
The lo1 Example (Part 1 of 5) The following is an example of a linear optimization
problem with one equality and two inequality constraints:
maximize 3x
1
+ 1x
2
+ 5x
3
+ 1x
4
subject to 3x
1
+ 1x
2
+ 2x
3
= 30,
2x
1
+ 1x
2
+ 3x
3
+ 1x
4
15,
2x
2
+ 3x
4
25,
(2.2)
having the bounds
0 x
1
,
0 x
2
10,
0 x
3
,
0 x
4
.
(2.3)
This is easily programmed in R as shown in Figure 4. The rst line overwrites any previous
denitions of the variable lo1, preparing for the new problem description. The problem is
then dened and nally solved on the last line.
7
Check out: mosek.com Documentation Optimization tools manual Modeling Linear
optimization.
17
2.2 Linear Programming A GUIDED TOUR
Figure 4: Linear Optimization (lo1)
lo1 <- list()
lo1$sense <- "max"
lo1$c <- c(3,1,5,1)
lo1$A <- Matrix(c(3,1,2,0,
2,1,3,1,
0,2,0,3), nrow=3, byrow=TRUE , sparse=TRUE)
blc <- c(30,15,-Inf)
buc <- c(30,Inf ,25); lo1$bc <- rbind(blc , buc);
blx <- c(0,0,0,0)
bux <- c(Inf ,10,Inf ,Inf); lo1$bx <- rbind(blx , bux);
r <- mosek(lo1)
Notice how the R value Inf is used in both the constraint bounds (blc and buc) and the
variable upper bound (bux), to avoid the specication of an actual bound. If this example
does not work you should go back to Section 1 on setting up the interface.

From this example the input arguments for the linear program (2.1) follows easily (refer
to the denitions of input arguments in Figure 2, Section 2.1).
Objective The string sense is the objective goal and could be either minimize,
min, maximize or max. The dense numeric vector c species the coecients
in front of the variables in the linear objective function, and the optional constant
scalar c0 (reads: c zero) is a constant in the objective corresponding to c
0
in
problem (2.1), that will be assumed zero if not specied.
Constraint Matrix The sparse matrix A is the constraint matrix of the problem
with the constraint coecients written row-wise. Notice that for larger problems
it may be more convenient to dene an empty sparse matrix and specify the non-
zero elements one at a time A(i, j) = a
ij
, rather than writing out the full matrix
as done in the lo1 example. E.g. Matrix(0,nrow=30,ncol=50,sparse=TRUE).
Bounds The constraint bounds bc with rows blc (constraint lower bound) and buc
(constraint upper bound), as well as the variable bounds bx with rows blx (variable
lower bound) and bux (variable upper bound), are both given as dense numeric
matrices. These are equivalent to the bounds of problem (2.1), namely l
c
, u
c
, l
x
and u
x
.
18
2.2 Linear Programming A GUIDED TOUR
Errors, warnings and response codes
If the mosek function is executed with a problem description as input, a log of the interface
and optimization process is printed to the screen revealing any errors or warnings the
process may have encountered. As a rule of thumb, errors will be given when a crucial part
of the problem description is missing, or when an input argument is set to a value that
does not make sense or is formatted incorrectly. Warnings on the other hand will be given
if some ignorable part of the problem has an empty denition (NULL, NA or NaN), or if the
interface has to convert or otherwise guess on an interpretation of input on a non-standard
form. Errors will always interrupt the optimization process whereas warnings will not.
Since warnings can hold valuable debugging information and may be important to notice,
they are both printed in the log at the time they occurred and later summarized just before
the interface returns.
Error messages works ne when you are interacting with the interface, but in automated
optimization frameworks they are not easily handled. This is why a response is always
returned as part of the result, when calling a function that may produce errors (see e.g.
Figure 2). The response is a list containing a code and msg. When an error happens inside
a function call to the MOSEK optimization library, the code is the response code returned
by the function call
8
and msg is the corresponding error message. When an error happens
within the interface, the code equals NaN and the msg is the error message which, in case
of unexpected execution paths, may require technical knowledge to understand. When no
errors are encountered, the code is zero, but beware that comparison operators such as
== does not handle NaN very good. Using identical may be a better alternative.
NB! The interface may return only partially constructed output.
(always check the response code; e.g. success <- identical(response_code, 0))
Interpreting the solution
The default optimizer for linear problems is the interior-point algorithm in MOSEK which
returns two solutions in the output structure. The interior-point solution (called itr) is the
solution found directly by the interior-point algorithm. The basic solution (called bas) is a
vertex solution derived from the values of itr, and could for instance be used to hot-start
the simplex method if small changes was applied at some point later. If another optimizer
using the simplex method was selected instead of the interior-point algorithm, the bas
solution would have been found directly and the itr solution would not exist.
8
Check out: mosek.com Documentation C API manual Response codes.
19
2.2 Linear Programming A GUIDED TOUR
The lo1 Example (Part 2 of 5) The lo1 example was solved using the default
optimizer (the interior-point algorithm) and contains two solutions: the interior-point (itr)
and the basic solution (bas) partly shown here.
As seen in Figure 5 and Figure 6 the solution space of the problem was not empty (as it
is primal feasible) and the objective was not unbounded (as it is dual feasible). In addition
the optimizer was able to identify the optimal solution.
Figure 5: Primal Solution I (lo1)
r$sol$itr
{
$solsta
[1] "OPTIMAL"
$prosta
[1] "PRIMAL_AND_DUAL_FEASIBLE"
$skc
[1] "EQ" "SB" "UL"
$skx
[1] "LL" "LL" "SB" "SB"
$skn
character (0)
$xc
[1] 30.00000 53.33333 25.00000
$xx
[1] 1.044111e-08 2.856068e-08 1.500000e+01 8.333333e+00
...
}
Figure 6: Primal Solution II (lo1)
r$sol$bas
{
$solsta
[1] "OPTIMAL"
$prosta
[1] "PRIMAL_AND_DUAL_FEASIBLE"
$skc
[1] "EQ" "BS" "UL"
$skx
[1] "LL" "LL" "BS" "BS"
$xc
[1] 30.00000 53.33333 25.00000
$xx
[1] 0.000000 0.000000 15.000000 8.333333
...
}
Notice that the basic solution bas is likely to have a higher numerical accuracy than the
interior-point solution itr as is the case in this example considering the xx variables.

20
2.2 Linear Programming A GUIDED TOUR
The solution you receive from the interface will contain the primal variable x (called xx)
and the activity of each constraint, xc, dened by x
c
= Ax. From the solution status
(called solsta) it can be seen how good this solution is, e.g. optimal, nearly optimal,
feasible or infeasible. If the solution status is not as you expected, it might be that the
problem is either ill-posed, infeasible or unbounded (dual infeasible). This can be read
from the problem status (called prosta). The solution and problem status are based on
certicates found by the MOSEK optimization library
9
, and should always be veried
before the returned solution values are used (see Section 2.2.5).
The solution also contains status key vectors for both variables and constraints (called skx
and skc). The variable skn is not useful for linear programming problems. Each status
key will be one of the following two-character strings.
BS : Basic
In basic (bas) solutions: The constraint or variable belongs to the basis of the
corresponding simplex tableau.
SB : Super Basic
In interior-point (itr) and integer (int) solutions: The activity of a constraint or variable
is in between its bounds.
LL : Lower Level
The constraint or variable is at its lower bound.
UL : Upper Level
The constraint or variable is at its upper bound.
EQ : Equality
The constraint or variable is at its xed value (equal lower and upper bound).
UN : Unknown
The status of the constraint or variable could not be determined (in practice never
returned by MOSEK).
In addition to the primal variables just presented, the returned solutions also contains dual
variables not shown here. The dual variables can be used for sensitivity analysis of the
problem parameters and are related to the dual problem explained in Section 2.2.2.
9
More details on problem and solution status keys available at:
mosek.com Documentation Optimization tools manual Symbolic constants reference.
21
2.2 Linear Programming A GUIDED TOUR
2.2.2 Duality
The dual problem corresponding to the primal problem (2.1) dened in Section 2.2.1, is
shown below in (2.4). Notice that the coecients of the dual problem is the same as those
used in the primal problem. Matrix A for example is still the constraint matrix of the
primal problem, merely transposed in the dual problem.
In addition, the dual problem have dual variables for each lower and upper, constraint
and variable bound in the primal problem: s
c
l
R
m
, s
c
u
R
m
, s
x
l
R
n
and s
x
u
R
n
(the
latter being the dual variable of the upper variable bound).
The dual problem is given by
maximize (l
c
)
T
s
c
l
(u
c
)
T
s
c
u
+ (l
x
)
T
s
x
l
(u
x
)
T
s
x
u
+ c
0
subject to A
T
(s
c
l
s
c
u
) + s
x
l
s
x
u
= c ,
s
c
l
, s
c
u
, s
x
l
, s
x
u
0 .
(2.4)
Recall that equality constraints had to be specied using two inequalities (with l
c
= u
c
)
by denition of the primal problem (2.1). This means that an equality constraint will have
two dual variables instead of just one. If the user wants to calculate the one dual variable,
as it would have been if equality constraints could be specied directly, then this is given
by s
c
l
s
c
u
. However, it is not always recommended to do so, as it is often easier to stay
with the same problem formulation and do all calculations directly on that.
The lo1 Example (Part 3 of 5) The part of the solution to the lo1 example that
was previously omitted, is now shown in Figure 7 and 8. The dual variables slc, suc, slx
and sux corresponds naturally to s
c
l
, s
c
u
, s
x
l
and s
x
u
in the dual problem. The variable snx
is not useful for linear programming problems.
Looking at the denition of the lo1 problem (2.2), the rst constraint is an equality,
the second is an lower bound, and the third is an upper bound. The dual variable of
the inequalities should just be read from slc and suc respectively, while for the equality
constraint, having two dual variables, you could also look at the combined lower minus
upper constraint dual variable (slc - suc), which in this case would give you a dual value
of 2.5.
22
2.2 Linear Programming A GUIDED TOUR
Figure 7: Dual Solution I (lo1)
r$sol$itr
{
$solsta
[1] "OPTIMAL"
$prosta
[1] "PRIMAL_AND_DUAL_FEASIBLE"
...
$slc
[1] 0.000000e+00 1.557535e-09 0.000000e+00
$suc
[1] -2.5000000 0.0000000 -0.3333333
$slx
[1] -4.500000e+00 -2.166667e+00 -4.982961e-09 -1.027032e-08
$sux
[1] 0.0000e+00 5.5856e-10 0.0000e+00 0.0000e+00
$snx
[1] 0 0 0 0
}
Figure 8: Dual Solution II (lo1)
r$sol$bas
{
$solsta
[1] "OPTIMAL"
$prosta
[1] "PRIMAL_AND_DUAL_FEASIBLE"
...
$slc
[1] 0 0 0
$suc
[1] -2.5000000 0.0000000 -0.3333333
$slx
[1] -4.500000 -2.166667 0.000000 0.000000
$sux
[1] 0 0 0 0
}
Notice that the basic solution bas is likely to have a higher numerical accuracy than the
interior-point solution itr as is the case here.

2.2.3 Switching Optimizer

The integer parameter
10
OPTIMIZER controls which optimizer to use within the MOSEK
optimization library to solve the specied problem. The default value of this parameter is
10
Check out Section 2.6 for more details on parameter settings.
23
2.2 Linear Programming A GUIDED TOUR
the enum reference-string OPTIMIZER_FREE which imply that MOSEK should choose
an optimizer on its own. Currently MOSEK always selects the interior-point algorithm
for linear programming problems which performs especially well for large optimization
problems, but for small to medium sized problems it might sometimes be benecial to
switch over to the simplex method.
To solve a linear programming problem using another user-specied optimizer, the
OPTIMIZER parameter can be set to one of the following reference-strings
11
:
OPTIMIZER_FREE
The default parameter setting discussed above.
OPTIMIZER_INTPNT
The interior-point algorithm.
OPTIMIZER_FREE_SIMPLEX
The simplex method on either the primal or dual problem (MOSEK selects).
OPTIMIZER_PRIMAL_SIMPLEX
The simplex method on the primal problem.
OPTIMIZER_DUAL_SIMPLEX
The simplex method on the dual problem.
The lo1 Example (Part 4 of 5) In this example we shall try to solve the lo1 example
from Section 2.2.1 using the primal simplex method. This is specied by setting the
integer parameter to the reference-string OPTIMIZER_PRIMAL_SIMPLEX as shown
in Figure 9, adding a single line to the lo1 denition from earlier.
Figure 9: Selecting the primal simplex method
lo1 <- list()
lo1$sense <- "max"
lo1$c <- c(3,1,5,1)
lo1$A <- Matrix(c(3,1,2,0,
2,1,3,1,
0,2,0,3), nrow=3, byrow=TRUE , sparse=TRUE)
blc <- c(30,15,-Inf)
buc <- c(30,Inf ,25); lo1$bc <- rbind(blc , buc);
blx <- c(30,15,-Inf)
bux <- c(30,Inf ,25); lo1$bx <- rbind(blx , bux);
lo1$iparam <- list(OPTIMIZER =" OPTIMIZER_PRIMAL_SIMPLEX ");
r <- mosek(lo1)
11
More values for the MSK_IPAR_OPTIMIZER parameter available at:
mosek.com Documentation Optimization tools manual Parameter reference.
24
2.2 Linear Programming A GUIDED TOUR
The output only contains the optimal basic solution bas, which is equivalent to the basic
solution found by the interior-point algorithm in the previous lo1 examples. To verify
that the primal simplex method was actually the optimizer used for this problem, we can
check the log printed to the screen and shown in Figure 10.
Figure 10: The log veries the choice of optimizer
...
Optimizer started.
Simplex optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator - tries : 0 time : 0.00
Eliminator - elim s : 0
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Primal simplex optimizer started.
Primal simplex optimizer setup started.
Primal simplex optimizer setup terminated.
...

2.2.4 Hot-starting
Hot-starting (also known as warm-starting) is a way to make the optimization process
aware of a point in the solution space which, depending on the quality of it (feasibility
and closeness to the optimal solution), can increase the speed of optimization. In linear
programming it is typically used when you know the optimal solution to a similar problem
with only few small changes to the constraints and objective. In these cases it is assumed
that the next optimal solution is nearby in the solution space, and thus it would also makes
sense to switch to the simplex optimizers excellent for small changes to the set of basic
variables - even on large problems. In fact, currently, the user will have to use one of the
simplex optimizers for hot-starting in linear programming, as the interior-point optimizer
in MOSEK cannot take advantage of initial solutions.
Simplex optimizers only look for the basic solution bas in the input argument $sol, and do
not consider the solution and problem statuses within. These may however be specied
anyway for the convenience of the user, and warnings will only be given if no useful
information could be given to the MOSEK optimizer despite the fact that $sol had a
non-empty denition and hot-starting was attempted.
25
2.2 Linear Programming A GUIDED TOUR
When adding a new variable
In column generation it is necessary to reoptimize the problem after one or more variables
have been added to the problem. Given a previous solution to the problem, the number of
basis changes would be small and we can hot-start using a simplex optimizer.
Assume that we would like to solve the problem
maximize 3x
1
+ 1x
2
+ 5x
3
+ 1x
4
1x
5
subject to 3x
1
+ 1x
2
+ 2x
3
2x
5
= 30,
2x
1
+ 1x
2
+ 3x
3
+ 1x
4
10x
5
15,
2x
2
+ 3x
4
+ 1x
5
25,
(2.5)
having the bounds
0 x
1
,
0 x
2
10,
0 x
3
,
0 x
4
,
0 x
5
,
(2.6)
which is equal to the lo1 problem (2.2), except that a new variable x
5
has been added.
To hot-start from the previous solution of Figure 4, which is still primal feasible, we can
expand the problem description and include x
5
as shown.
Figure 11: Hot-starting when a new variable is added
# Define lo1 and obtain the solution r (not shown)
lo1_backup <- lo1
# Append the new variable to the problem:
lo1$c <- c(lo1$c , -1)
lo1$bx <- cbind(lo1$bx , c(0,Inf))
lo1$A <- cBind(lo1$A , c(-2,-10,1))
# Extend and reuse the old basis solution:
oldbas <- r$sol$bas
oldbas$skx <- c(oldbas$skx , LL )
oldbas$xx <- c(oldbas$xx , 0)
oldbas$slx <- c(oldbas$slx , 0)
oldbas$sux <- c(oldbas$sux , 0)
# Hot -start the simplex optimizer:
lo1$iparam <- list(OPTIMIZER =" OPTIMIZER_PRIMAL_SIMPLEX ")
lo1$sol <- list(bas=oldbas)
r_var <- mosek(lo1)
26
2.2 Linear Programming A GUIDED TOUR
When xing a variable
In branch-and-bound methods for integer programming it is necessary to reoptimize the
problem after a variable has been xed to a value. From the solution of the lo1 problem
(Figure 4), we x the variable x
4
= 2, and hot-start using
Figure 12: Hot-starting when a variable has been xed
# Define lo1 and obtain the solution r (not shown)
lo1_backup <- lo1
# Fix the fourth variable in the problem
lo1$bx [,4] <- c(2,2)
# Reuse the old basis solution
oldbas <- r$sol$bas
# Hotstart the simplex optimizer
lo1$iparam <- list(OPTIMIZER =" OPTIMIZER_DUAL_SIMPLEX ")
lo1$sol <- list(bas=oldbas)
r_fix <- mosek(lo1)
When adding a new constraint
In cutting plane algorithms it is necessary to reoptimize the problem after one or more
constraints have been added to the problem. From the solution of the lo1 problem (Figure
4), we add the constraint x
1
+ x
2
2, and hot-start using
Figure 13: Hot-starting when a constraint has been added
# Define lo1 and obtain the solution r (not shown)
lo1_backup <- lo1
# Append the new constraint to the problem:
lo1$bc <- cbind(lo1$bc , c(2,Inf))
lo1$A <- rBind(lo1$A , c(1,1,0,0))
# Extend and reuse the old basis solution:
oldbas <- r$sol$bas
oldbas$skc <- c(oldbas$skc , LL )
oldbas$xc <- c(oldbas$xc , 2)
oldbas$slc <- c(oldbas$slc , 0)
oldbas$suc <- c(oldbas$suc , 0)
# Hot -start the simplex optimizer:
lo1$iparam <- list(OPTIMIZER =" OPTIMIZER_DUAL_SIMPLEX ")
lo1$sol <- list(bas=oldbas)
r_con <- mosek(lo1)
27
2.2 Linear Programming A GUIDED TOUR
Using numerical values to represent status keys
In the previous examples the status keys of constraints and variables were all dened as
two-character string codes. Although this makes the status keys easy to read, it might
sometimes be easier to work with numerical values. For this reason we now demonstrate
how to achieve this with the R-to-MOSEK interface. The explanation of status keys can
be found on page 21.
Figure 14: Hot-starting using an initial guess
# Define lo1 (not shown)
# Define the status key string array
stkeys <- c("BS","SB","LL","UL","EQ","UN")
# Try to guess the optimal status keys
mybas <- list()
mybas$skc = stkeys[c(5,1,4)]
mybas$skx = stkeys[c(3,3,1,1)]
# Hot -start the simplex optimizer
lo1$iparam <- list(OPTIMIZER =" OPTIMIZER_PRIMAL_SIMPLEX ")
lo1$sol <- list(bas=mybas)
r_guess <- mosek(lo1)
So basically stkeys[[idx]] will return the status key of index idx, and can be vectorized
as stkeys[idxvec] for a vector of indexes idxvec. Going in the opposite direction from
status keys to indexes requires a bit more work. For this you can use match(str,stkeys)
to nd the index of a status key str, and equivalently match(strvec,stkeys) for a vector
of status keys strvec.
28
2.2 Linear Programming A GUIDED TOUR
2.2.5 Complete Best Practice example
Linear programs, as well as other optimization models, are often solved as part of a larger
framework where the solutions are not just printed on the screen, but instead given as
input to other scripts and programs. Such frameworks include mathematical constructions
such as branch-and-price, but also include programs with internal optimization models
hidden from the end user. In these cases, special attention must be given to the handling
of function calls and return values.
Catch execution errors
From within the Rmosek package, execution errors similar to the ones generated by the
built-in stop function, part of the R language denition, may be provoked. This typically
happens when the number of input arguments does not match a valid calling convention,
or when one of the input arguments can not be evaluated. Interface errors within the R
API or Matrix package could, however, also generate these kinds of errors.
When execution errors are provoked in some function, the control is returned to the
outer-most scope (the global environment) and no values are returned from the function
call. That is, when calling res <- some_function(...) and an execution error is
provoked, the variable res will remain unaected and could host old obsolete values.
Luckily, execution errors can be caught with the built-in try function which takes a
function call as input argument. The result of this try function (say res), will either
be the returned value of the function call or some error class. The error class inherits
from "try-error" and can be distinguished from normal output by the boolean result of
inherits(res, "try-error").
Inspect the reponse code
The response code may either be a zero (success), a positive integer (error in the MOSEK
optimization library) or simply NaN (error in the Rmosek interface). The list of response
codes that can be returned by the MOSEK optimization library, are similar to those
explained online
12
. As explained in Errors, warnings and response codes in Section
2.2.1, always use the built-in identical function to compare reponse codes.
When an error is encountered, the interface will still try to extract and return a solution.
If for instance the optimization procedure ran out of memory or time, or was interrupted
by the keyboard sequence <CTRL> + <C> (see e.g. Section 2.8.2), a good solution may
have been identied anyway. However, if the extraction of the solution was what caused the
12
Check out: mosek.com Documentation C API manual Response codes.
29
2.2 Linear Programming A GUIDED TOUR
error in the rst place, the returned solution may only be partially constructed. Ultimately,
no guarantees about the availability of variables can be given when the response code is
not zero.
In case of a non-zero response code, the availability of a variable within a result res,
can be tested with an expression similar to !inherits(try({res$sol$itr$solsta})),
"try-error"), which is TRUE when a solution status to a interior-point solution in the
result has been dened. Notice that a scope has been dened in the input argument of
the try function using { and }. Within this scope, several variables can be tested for
existence, or alternatively, entire blocks of code may be inserted and attempted evaluated.
Inspect the solution status
The solution status is based on certicates found by the MOSEK optimization library, and
classify the returned solution. In the list below, the two rst status keys also exists for
the dual case with PRIMAL replaced by DUAL. Also, all these seven status keys (excluding
UNKNOWN) has an nearly equivalent status key with NEAR_ added as a prex.
PRIMAL_INFEASIBLE_CER (linear problems only)
The returned data contains a certicate that the problem is primal infeasible.
PRIMAL_FEASIBLE
The returned data contains primal feasible solution.
PRIMAL_AND_DUAL_FEASIBLE
The returned data contains a primal and dual feasible solution, but is not optimal.
OPTIMAL
The returned data contains an optimal solution.
INTEGER_OPTIMAL (integer problems only)
The returned data contains an integer optimal solution.
UNKNOWN
The returned data should not be used.
A response code of zero is not enough to conclude that a solution can be extracted from
the returned data. In particular, the returned data may be a certicate of primal or dual
infeasibility instead of a solution to the problem. Moreover, the returned primal variables
may not contain any useful information if the solution status is e.g. DUAL_FEASIBLE. This
also holds true for dual variables if the solution status is e.g. PRIMAL_FEASIBLE. Thus it is
important to check and respond to the solution status before using the solution variables.
30
2.2 Linear Programming A GUIDED TOUR
Inspect the problem status
The problem status is based on certicates found by the MOSEK optimization library,
and classify the specied problem description. In the list below, the three rst status keys
also exists for the dual case with PRIMAL replaced by DUAL, and with PRIMAL replaced by
PRIMAL_AND_DUAL.
PRIMAL_INFEASIBLE
The problem description is primal infeasible.
NEAR_PRIMAL_FEASIBLE
The problem description satises a relaxed primal feasibility criterion.
PRIMAL_FEASIBLE
The problem description is primal feasible.
ILL_POSED (non-linear problems only)
The problem description has an unstable formulation.
Regarding the usefulness of the returned data, the solution status often tells the whole
story. For mixed integer programs (see Section 2.5), there is however no certicates for
infeasibility and this status have to be read from the problem status. Also, if the solution
status is UNKNOWN, the problem status may still contain valuable information. Furthermore,
for non-linear problems, it is a good idea to verify that the problem status is not ILL_POSED.
The lo1 Example (Part 5 of 5) In this example we dene a function which shall
try to solve the lo1 example from Section 2.2.1, and return the variable activities of the
basic solution. The function is completely silent except for the explicit cat-commands,
and will not fail to return a usable result unless explicitly stopped by a stop-command.
Furthermore, the double-typed parameter OPTIMIZE_MAX_TIME (see e.g. the FAQ,
Section 2.8.3) can be controlled by the input argument maxtime. If zero, MOSEK will have
no time to identify a useful solution. The function is shown in Figure 15.
As seen, the verbosity is set to zero with the option verbose to specify that the interface
should be completely silent (see the FAQ, Section 2.8.1). Also, any execution errors that
may be thrown by the mosek function is caught. The function then veries the response
code, and chooses to continue even this is not zero (you may choose to do otherwise). It
then extracts the basic solution and, at the same time, evaluate whether all the variables
that we are going to use are dened. We do not bother to check if the problem status is
ill-posed, as lo1 is a linear program, but instead continue to the solution status which is
only accepted if it is close to optimal (again, you may choose to do otherwise). Finally the
solution variable activities of the solution is returned.
31
2.2 Linear Programming A GUIDED TOUR
Figure 15: Complete Best Practice lo1 example
get_lo1_solution_variables <- function(maxtime) {
lo1 <- list(sense="max", c=c(3,1,5,1))
lo1$A <- Matrix(c(3,1,2,0,2,1,3,1,0,2,0,3),
nrow=3, byrow=TRUE , sparse=TRUE)
lo1$bc <- rbind(blc=c(30,15,-Inf), buc=c(30,Inf ,25));
lo1$bx <- rbind(blx=c(0,0,0,0), bux=c(Inf ,10,Inf ,Inf));
lo1$dparam <- list(OPTIMIZER_MAX_TIME=maxtime)
r <- try(mosek(lo1 , list(verbose =0)), silent=TRUE)
if (inherits(r, "try -error ")) {
stop(" Rmosek failed somehow !")
}
if (! identical(r$response$code , 0)) {
cat(paste ("**" , "Response code:", r$response$code , "\n"))
cat(paste ("**" , r$response$msg , "\n"))
cat(" Trying to continue ..\n")
}
isdef <- try({
rbas <- r$sol$bas;
rbas$solsta; rbas$prosta; rbas$xx;
}, silent=TRUE)
if (inherits(isdef , "try -error ")) {
stop(" Basic solution was incomplete !")
}
switch(rbas$solsta ,
OPTIMAL = {
cat("The solution was optimal , I am happy!\n")
},
NEAR_OPTIMAL = {
cat("The solution was close to optimal , very good ..\n")
},
#OTHERWISE:
{
cat(paste ("**" , "Solution status:", rbas$solsta , "\n"))
cat(paste ("**" , "Problem status:", rbas$prosta , "\n"))
stop(" Solution could not be accepted !")
}
)
return(rbas$xx)
}

32
2.3 Conic Quadratic Programming A GUIDED TOUR
2.3 Conic Quadratic Programming
2.3.1 Solving CQP problems
Conic quadratic programming
13
(also known as Second-order cone programming) is a
generalization of linear, quadratic and all-quadratic programming. Conic quadratic
programs can be written as shown below, and pose the properties of strong duality under
Slaters condition. This is the primal problem.
minimize c
T
x + c
0
subject to l
c
Ax u
c
,
l
x
x u
x
,
x (.
(2.7)
The convex cone ( can be written as the Cartesian product over a nite set of convex cones
( = (
1
(
p
, which basically means that the variables can be partitioned into subsets
of variables belonging to dierent cones. In principle this also means that each variable can
only belong to one cone, but in practice we can dene several duplicates x
i
of x
i
belonging
to dierent cones and connected by x
i
= x
i
in the linear constraints of (2.7).
The MOSEK optimization library currently allows three types of convex cones: The R-
cone, the quadratic cone and the rotated quadratic cone. The R-cone contains the full
set of real numbers and is the default cone in this interface for variables with no other
specication. Notice that if all variables belonged to this cone the problem would reduce
to a linear programming problem. The quadratic cone is dened by
(
t
=
_
_
_
x R
nt
: x
1

_
nt

j=2
x
2
j
_
_
_
(2.8)
for which the indexes shown here refer only to the subset of variables belonging to the
cone. Similarly the rotated quadratic cone is given by
(
t
=
_
_
_
x R
nt
: 2x
1
x
2

nt

j=3
x
2
j
, x
1
0, x
2
0
_
_
_
. (2.9)
These denitions may seem restrictive, but can model a large number of problems as shown
by the transformations of Appendix B.
13
Check out: mosek.com Documentation Optimization tools manual Modeling Conic
optimization.
33
2.3 Conic Quadratic Programming A GUIDED TOUR
The cqo1 Example (Part 1 of 3) The following is an example of a conic optimization
problem with one linear constraint, non-negative variables and two cones:
minimize x
4
+ x
5
+ x
6
subject to x
1
+ x
2
+ 2x
3
= 1 ,
x
1
, x
2
, x
3
0 ,
x
4

_
x
2
1
+ x
2
2
,
2x
5
x
6
x
2
3
.
(2.10)
The rst cone is of the quadratic cone type (MSK_CT_QUAD), while the second is of the
rotated quadratic cone type (MSK_CT_RQUAD.) The subindexes of the variables used
to dene these cones follow naturally as seen in the following R code.
Figure 16: Conic Quadratic Optimization (cqo1)
cqo1 <- list(sense = "min")
cqo1$c <- c(0,0,0,1,1,1)
cqo1$A <- Matrix(c(1,1,2,0,0,0),
nrow=1, byrow=TRUE , sparse=TRUE)
cqo1$bc <- rbind(blc = 1, buc = 1)
cqo1$bx <- rbind(blx = c(0,0,0,-Inf ,-Inf ,-Inf),
bux = rep(Inf ,6))
cqo1$cones <- cbind(
list("QUAD", c(4,1,2)),
list(" RQUAD", c(5,6,3))
)
rownames(cqo1$cones) <- c("type","sub ");
r <- mosek(cqo1)

From this example the input arguments for a conic program (2.7) follow easily (refer to
Figure 2, Section 2.1). The objective function, the linear constraints and variable bounds
should all be specied as for linear programs (see Section 2.2), and the only addition to
this is the quadratic cones specied in the list-typed matrix cones.
The cones matrix has a column for each cone, and a row for each descriptive element.
The rst row called type, should specify the cone type in a string, being either quadratic
QUAD or rotated quadratic RQUAD. Notice that the MOSEK library cone type prex
MSK_CT_ is optional. The second row called sub, should specify the subset of variables
belonging to the cone in a numeric vector - and the ordering does matter! The ith element
of sub will be the index of the variable referred by x
i
, in the cone denitions (2.8) and (2.9).
As an example, the rotated quadratic cone with subindexes c(4,6,2,3) would dene the
cone
(
t
=
_
x R
4
: 2x
4
x
6
x
2
2
+ x
2
3
, x
4
0, x
6
0
_
. (2.11)
34
2.3 Conic Quadratic Programming A GUIDED TOUR
Figure 16 showed a simple way to specify cones given an explicit representation. In many
practical cases, however, cones are more conveniently specied in chunks or within a loop.
For this purpose, preallocation should always be preferred as shown here.
Figure 17: Preallocating cones (cqo1)
NUMCONES <- 2
cqo1$cones <- matrix(list(), nrow=2, ncol=NUMCONES)
rownames(cqo1$cones) <- c("type","sub")
cqo1$cones [,1] <- list("QUAD", c(4,1,2))
cqo1$cones [,2] <- list("RQUAD", c(5,6,3))

Errors, warnings and response codes (as in Linear Programming)

If the mosek function is executed with a problem description as input, a log of the interface
and optimization process is printed to the screen revealing any errors or warnings the
process may have encountered. As a rule of thumb, errors will be given when a crucial part
of the problem description is missing, or when an input argument is set to a value that
does not make sense or is formatted incorrectly. Warnings on the other hand will be given
if some ignorable part of the problem has an empty denition (NULL, NA or NaN), or if the
interface has to convert or otherwise guess on an interpretation of input on a non-standard
form. Errors will always interrupt the optimization process whereas warnings will not.
Since warnings can hold valuable debugging information and may be important to notice,
they are both printed in the log at the time they occurred and later summarized just before
the interface returns.
Error messages works ne when you are interacting with the interface, but in automated
optimization frameworks they are not easily handled. This is why a response is always
returned as part of the result, when calling a function that may produce errors (see e.g.
Figure 2). The response is a list containing a code and msg. When an error happens inside
a function call to the MOSEK optimization library, the code is the response code returned
by the function call
14
and msg is the corresponding error message. When an error happens
within the interface, the code equals NaN and the msg is the error message which, in case
of unexpected execution paths, may require technical knowledge to understand. When no
errors are encountered, the code is zero, but beware that comparison operators such as
== does not handle NaN very good. Using identical may be a better alternative.
NB! The interface may return only partially constructed output.
(always check the response code; e.g. success <- identical(response_code, 0))
14
Check out: mosek.com Documentation C API manual Response codes.
35
2.3 Conic Quadratic Programming A GUIDED TOUR
Interpreting the solution
The default optimizer for conic quadratic problems is the interior-point algorithm which
only returns the interior-point solution (called itr).
The cqo1 Example (Part 2 of 3) The cqo example was solved using the default
optimizer (the interior-point algorithm) and contains the interior-point solution partly
presented here.
As seen in Figure 18 the solution space of the problem was not empty (as it is primal
feasible) and the problem was not unbounded (as it is dual feasible). In addition the
optimizer was able to identify one of the optimal solutions.
Figure 18: Primal Solution (cqo1)
r$sol$itr
{
$solsta
[1] "OPTIMAL"
$prosta
[1] "PRIMAL_AND_DUAL_FEASIBLE"
$skc
[1] "EQ"
$skx
[1] "SB" "SB" "SB" "SB" "SB" "SB"
$skn
[1] "UL" "UL"
$xc
[1] 1
$xx
[1] 0.2609205 0.2609205 0.2390795 0.3689974 0.1690547 0.1690547
...
}

The solution you receive from the interface will contain the primal variable x (called xx)
and the activity of each constraint, xc, dened by x
c
= Ax. From the solution status
(called solsta) it can be seen how good this solution is, e.g. optimal, nearly optimal,
feasible or infeasible. If the solution status is not as you expected, it might be that the
problem is either ill-posed, infeasible or unbounded (dual infeasible). This can be read
from the problem status (called prosta). The solution and problem status are based on
certicates found by the MOSEK optimization library
15
, and should always be veried
before the returned solution values are used (see Section 2.2.5).
15
More details on problem and solution status keys available at:
mosek.com Documentation Optimization tools manual Symbolic constants reference.
36
2.3 Conic Quadratic Programming A GUIDED TOUR
The solution also contains status key vectors for both variables, linear constraints and
conic constraints (called skx, skc and skn). Each status key will be one of the following
two-character strings.
BS : Basic
In basic (bas) solutions: The constraint or variable belongs to the basis of the
corresponding simplex tableau.
SB : Super Basic
In interior-point (itr) and integer (int) solutions: The constraint or variable is in
between its bounds.
LL : Lower Level
The constraint or variable is at its lower bound.
UL : Upper Level
The constraint or variable is at its upper bound.
EQ : Equality
The constraint or variable is at its xed value (equal lower and upper bound).
UN : Unknown
The status of the constraint or variable could not be determined.
In addition to the primal variables just presented, the returned solutions also contain dual
variables not shown here. The dual variables can be used for sensitivity analysis of the
problem parameters and are related to the dual problem explained in Section 2.3.2.
37
2.3 Conic Quadratic Programming A GUIDED TOUR
2.3.2 Duality
The dual problem corresponding to the primal conic quadratic problem (2.7) dened in
Section 2.3.1, is shown below in (2.12). Notice that the coecients of the dual problem is
the same as those used in the primal problem. Matrix A for example is still the constraint
matrix of the primal problem, merely transposed in the dual problem formulation.
The dual problem is very similar to that of linear programming (Section 2.2.2), except
for the added variable s
x
n
belonging to the dual cone of ( given by (

.
maximize (l
c
)
T
s
c
l
(u
c
)
T
s
c
u
+ (l
x
)
T
s
x
l
(u
x
)
T
s
x
u
+ c
0
subject to A
T
(s
c
l
s
c
u
) + s
x
l
s
x
u
+ s
x
n
= c ,
s
c
l
, s
c
u
, s
x
l
, s
x
u
0 ,
s
x
n
(

.
(2.12)
The dual cone (

always has the same number of variables as its primal (, and is in fact
easily expressed for the three convex cone types supported by MOSEK. For the R-cone its
dual is the single point in space with dual variables s
x
n
all zero, such that (2.12) coincides
with the dual linear program (2.4), when all primal variables belong to the R-cone.
Just as easily it holds true that both the quadratic and rotated quadratic cones are self
dual such that ( = (

. These facts ease the formulation of dual conic problems and have
been exploited by the MOSEK optimization library for fast computations.
The cqo1 Example (Part 3 of 3) The part of the solutions to the cqo1 example
that was previously omitted, is now shown in Figure 19. The dual variables slc, suc, slx,
sux and snx correspond naturally to s
c
l
, s
c
u
, s
x
l
, s
x
u
and s
x
n
in the dual problem.
Looking at the denition of the cqo1 problem (2.10), the rst and only constraint is an
equality constraint why its dual variables can either be read individually from its implicit
lower an upper bound, slc and suc, or from the combined lower minus upper constraint
dual variable (slc - suc). Notice that since none of the primal variables have upper bounds,
the corresponding dual variables sux are all xed to zeros. Further more, since all primal
variables belong to a self-dual quadratic cone, all of the snx variables can attain values
from the corresponding quadratic cones.
38
2.3 Conic Quadratic Programming A GUIDED TOUR
Figure 19: Dual Solution (cqo1)
r$sol$itr
{
$solsta
[1] "OPTIMAL"
$prosta
[1] "PRIMAL_AND_DUAL_FEASIBLE"
...
$slc
[1] 0.7071068
$suc
[1] 0
$slx
[1] 2.582619e-09 2.582619e-09 3.845566e-09 0.000000e+00 0.000000e+00
[6] 0.000000e+00
$sux
[1] 0 0 0 0 0 0
$snx
[1] -0.7071068 -0.7071068 -1.4142136 1.0000000 1.0000000 1.0000000
}

39
2.3 Conic Quadratic Programming A GUIDED TOUR
2.3.3 Notes on Quadratic Programming (QP/QCQP)
Quadratic programming, or more generally all-quadratic programming, is a smaller subset
of conic quadratic programming (CQP) that can easily be handled and solved using modern
CQP solvers. The primal problem of a quadratic program can be stated as shown below
in (2.13), where matrices Q
0
, Q
1
, and columns a, b and c are assumed to have compliant
dimensions. Notice that this problem is convex, and thereby easily solvable, only if Q
0
and
Q
1
are both positive semidenite.
minimize 0.5 x
T
Q
0
x + c
T
x
subject to 0.5 x
T
Q
1
x + a
T
x b.
(2.13)
While quadratic terms are added directly to the objective function and constraint matrix
in quadratic programming, CQP formulations instead retain both a linear objective and a
linear constraint matrix, closely resembling a linear program. All non-linearities, such as
quadratic terms, are in this case formulated by themselves using quadratic cones. Many
types of non-linearities can be modeled using these quadratic cones as seen in Appendix B,
but in the case of (2.13), the result is two of the simplest possible rotated quadratic cones
- one for the x
T
Q
0
x term and one for the x
T
Q
1
x term.
CQP formulations also have the advantage that there are no conditions on when a model is
convex or not, because it always is. The transformation from quadratic programs to conic
quadratic programs is based on a matrix factorization requiring Q
0
and Q
1
to be positive
semidenite, which is exactly the convexity requirement for QCQPs. This means that
x
T
Q
0
x can be transformed only if x
T
Q
0
x is convex. In some sense, this transformation
can be seen as a model-strengthening preprocessing step that only has to be done once.
All further changes to the model can of course be made directly to the CQP.
The transformation of a quadratic program, whether it is QP or QCQP,
to a conic quadratic program (CQP), can be seen in Appendix B.1.
40
2.4 Semidenite Programming A GUIDED TOUR
2.4 Semidenite Programming
2.4.1 Solving SDP problems
Semidenite programming
16
is a generalization of linear and conic quadratic programming.
Semidenite programs can be written as shown below, and pose the properties of strong
duality under Slaters condition. This is the primal problem.
minimize c
T
x +

p
j=1
C
j
, X
j
) + c
0
subject to l
c
Ax +
_

p
j=1
A
1j
, X
j
)
.
.
.

p
j=1
A
mj
, X
j
)
_

_
u
c
,
l
x
x u
x
,
x (, X
j
o
+
r
j
for j = 1, . . . , p.
(2.14)
Cone ( is dened as in Section 2.3, but could just be the set of reals x R
n
. The problem
has p symmetric positive semidenite matrix variables X
j
o
+
r
j
of dimension r
j
r
j
, and
symmetric coecients C
j
o
r
j
and A
mj
o
r
j
. We use the standard notation for the
matrix inner product, i.e., for U, V R
mn
we have
U, V ) =
m

i=1
n

j=1
U
ij
V
ij
.
The sdo1 Example (Part 1 of 3) The following is an example of a semidenite
optimization problem with one linear objective and two linear constraints, with 9 scalar
variables from, respectively, a quadratic cone and a semidenite matrix:
minimize
_
_

_
2 1 0
1 2 1
0 1 2
_

_, X
1
_
+ x
1
subject to
_
_

_
1 0 0
0 1 0
0 0 1
_

_, X
1
_
+ x
1
= 1.0 ,
_
_

_
1 1 1
1 1 1
1 1 1
_

_, X
1
_
+ x
2
+ x
3
= 0.5 ,
x
1

_
x
2
2
+ x
2
3
,
X
1
_ 0 .
(2.15)
16
Check out: mosek.com Documentation Optimization tools manual Modeling Conic
optimization.
41
2.4 Semidenite Programming A GUIDED TOUR
The problem above corresponds to minimizing the objective:
2
_
[ X
1
]
11
+ [ X
1
]
22
+ [ X
1
]
33
_
+ 2
_
[ X
1
]
21
+ [ X
1
]
32
_
+ x
1
,
subject to the linear constraints
[ X
1
]
11
+ [ X
1
]
22
+ [ X
1
]
33
+ x
1
= 1.0 ,
[ X
1
]
11
+ [ X
1
]
22
+ [ X
1
]
33
+ 2
_
[ X
1
]
21
+ [ X
1
]
31
+ [ X
1
]
32
_
+ x
2
+ x
3
= 0.5 ,
and can be modeled in R as shown.
Figure 20: Semidenite Optimization (sdo1)
sdo1 <- list(sense="min")
sdo1$c <- c(1,0,0)
sdo1$A <- Matrix(c(1,0,0,
0,1,1), nrow=2, byrow=TRUE , sparse=TRUE)
sdo1$bc <- rBind(blc = c(1, 0.5), buc = c(1, 0.5))
sdo1$bx <- rBind(blx = rep(-Inf ,3), bux = rep(Inf ,3))
sdo1$cones <- cBind(list("quad", c(1 ,2 ,3)))
# One semidefinite matrix variable size 3x3:
N <- 3
sdo1$bardim <- c(N)
# Block triplet format specifying the lower triangular part
# of the symmetric coefficient matrix barc :
sdo1$barc$j <- c(1, 1, 1, 1, 1)
sdo1$barc$k <- c(1, 2, 3, 2, 3)
sdo1$barc$l <- c(1, 2, 3, 1, 2)
sdo1$barc$v <- c(2, 2, 2, 1, 1)
# Block triplet format specifying the lower triangular part
# of the symmetric coefficient matrix barA :
sdo1$barA$i <- c(1, 1, 1, 2, 2, 2, 2, 2, 2)
sdo1$barA$j <- c(1, 1, 1, 1, 1, 1, 1, 1, 1)
sdo1$barA$k <- c(1, 2, 3, 1, 2, 3, 2, 3, 3)
sdo1$barA$l <- c(1, 2, 3, 1, 2, 3, 1, 1, 2)
sdo1$barA$v <- c(1, 1, 1, 1, 1, 1, 1, 1, 1)
r <- mosek(sdo1)
barx <- 1.0 * bandSparse(N, k=0:(1-N), symm=TRUE)
barx@x <- r$sol$itr$barx [[1]]

42
2.4 Semidenite Programming A GUIDED TOUR
2.4.2 Duality
The dual problem corresponding to the primal semidenite problem (2.14), is shown below
in (2.16). Notice that the coecients of the dual problem is the same as those used in the
primal problem. Matrix A for example is still the constraint matrix of the primal problem,
merely transposed in the dual problem formulation.
The dual problem is very similar to that of conic quadratic programming (Section 2.12),
except for the added constraint and the p positive semidenite variables S
j
.
maximize (l
c
)
T
s
c
l
(u
c
)
T
s
c
u
+ (l
x
)
T
s
x
l
(u
x
)
T
s
x
u
+ c
0
subject to A
T
y + s
x
l
s
x
u
+ s
x
n
= c ,

m
i=1
y
i
A
ij
+ S
j
= C
j
j = 1, . . . , p,
y = s
c
l
s
c
u
and s
c
l
, s
c
u
, s
x
l
, s
x
u
0 ,
s
x
n
(

,
S
j
o
+
r
j
for j = 1, . . . , p .
(2.16)
43
2.5 Mixed Integer Programming A GUIDED TOUR
2.5 Mixed Integer Programming
2.5.1 Solving MILP and MICQP problems
Mixed Integer Programming is a common and very useful extension to both linear and
conic optimization problems, where one or more of the variables in the problem is required
only to attain integer values. For the user this will only require a modest change to the
model as these variables simply have to be pointed out, but to the underlying optimization
library the problem increases in complexity for every integer variable added
17
.
This happens because mixed integer optimization problems have to be solved using
continuous relaxations and branching strategies to force integrality. Consequently, the
running time of the optimization process will be highly dependent on the strength of the
continuous relaxation to the problem formulation - that is, how far from the optimal mixed
integer solution the problem formulation is when solved without the integrality requirement.
Some suggestions to reduce the solution time are thus:
Relax the termination criterion: In case the run time is not acceptable, the rst thing
to do is to relax the termination criterion (see Section 2.5.3)
Specify a good initial solution: In many cases a good feasible solution is either known
or easily computed using problem specic knowledge. If a good feasible solution is
known, it is usually worthwhile to use this as a starting point for the integer optimizer.
Improve the formulation: A mixed-integer optimization problem may be impossible
to solve in one form and quite easy in another form. However, it is beyond the scope
of this manual to discuss good formulations for mixed integer problems
18
.
A change that the user will notice when starting to enforce integrality is that the notion of a
dual problem is no longer dened for the problem at hand. This means that dual variables
will no longer be part of the solution to the optimization problem, and that only the primal
variables, constraint activities and problem/solution status reports, can be expected from
the output structure returned by the interface.
17
Check out: mosek.com Documentation Optimization tools manual The optimizer for mixed
integer problems.
18
See for instance: L. A. Wolsey, Integer programming, John Wiley and Sons, 1998
44
2.5 Mixed Integer Programming A GUIDED TOUR
The milo1 Example (Part 1 of 2) The way of requiring integer variables is the
same regardless of whether we solve linear or conic quadratic problems. The following is
an example of a simple mixed integer linear optimization problem, with two inequalities
and two non-negative integer variables:
maximize x
1
+ 0.64x
2
subject to 50x
1
+ 31x
2
250,
3x
1
2x
2
4,
x
1
, x
2
0 and integer.
(2.17)
This is easily programmed in R using the piece code shown in Figure 21, where x
1
and x
2
are pointed out as integer variables.
Figure 21: Mixed Integer Optimization (milo1)
milo1 <- list(sense = "max")
milo1$c <- c(1, 0.64)
milo1$A <- Matrix(c( 50, 31 ,
3, -2 ),
nrow = 2,
byrow = TRUE ,
sparse= TRUE)
milo1$bc <-
rbind(blc = c(-Inf ,-4),
buc = c(250,Inf))
milo1$bx <-
rbind(blx = c(0,0),
bux = c(Inf ,Inf))
milo1$intsub <- c(1,2)
r <- mosek(milo1)
Figure 22: Output object (milo1)
r$sol$int
{
$solsta
[1] "INTEGER_OPTIMAL"
$prosta
[1] "PRIMAL_FEASIBLE"
$skc
[1] "UL" "SB"
$skx
[1] "SB" "LL"
$xc
[1] 250 15
$xx
[1] 5 0
}

The input arguments follow those of a linear or conic program with the additional
identication of the integer variables (refer to Figure 2). The column vector intsub should
simply contain indexes to the subset of variables for which integrality is required. For
instance if x should be a binary {0,1}-variable, its index in the problem formulation should
be added to intsub, and its bounds 0 x 1 should be specied explicitly.
If executed correctly you should be able to see the log of the interface and optimization
process printed to the screen. The output structure shown in Figure 22, will only include
45
2.5 Mixed Integer Programming A GUIDED TOUR
an integer solution int, since we are no longer in the continuous domain for which the
interior-point algorithm operates. The structure also contains the problem status as well
as the solution status based on certicates found by the MOSEK optimization library
19
.
2.5.2 Hot-starting
Hot-starting (also known as warm-starting) is a way to make the optimization process
aware of a feasible point in the solution space which, depending on the quality of it
(closeness to the optimal solution), can increase the speed of optimization. In mixed
integer programming there are many ways to exploit a feasible solution and for anything
but small sized problems, it can only be recommended to let the optimizer know about a
feasible solution if one is available.
For many users the main advantage of hot-starting a mixed integer program will be the
increased performance, but others may also nd appreciation for the fact that the returned
solution can only be better (or in worst-case the same) as the solution fed in. This is
important in many applications where infeasible or low-quality solutions are not acceptable
even when time is short. Heuristics are thus combined with hot-started mixed integer
programs to yield a more reliable tool of optimization.
The milo1 Example (Part 2 of 2) For a small problem like the previously introduced
milo1, that can be solved to optimality without branching, it is not really useful to hot-
start. This example nevertheless illustrates the principle of how it can be done.
Figure 23: Hot-starting from initial guess (milo1)
# Define milo1 (not shown)
# Try to guess the optimal solution
myint <- list()
myint$xx <- c(5.0, 0.0)
# Hot -start the mixed integer optimizer
milo1$sol <- list(int=myint)
r <- mosek(milo1)

19
More details on problem and solution status keys available at:
mosek.com Documentation Optimization tools manual Symbolic constants reference.
46
2.5 Mixed Integer Programming A GUIDED TOUR
2.5.3 Termination criteria
A candidate solution, i.e. a solution to the continuously relaxed mixed integer problem, is
said to be integer feasible if the criterion
min
_
[x
j
[
_
[x
j
[
_
,
_
[x
j
[
_
[x
j
[
_
max (
1
,
2
[x
j
[) (2.18)
is satised for all variables in the problem, j J. Hence, such a solution is dened as
feasible to the mixed integer problem. Whenever the optimizer locates an integer feasible
solution it will check if the criterion
[z z[ max (
3
,
4
max (1, [z[)) (2.19)
is satised. If this is the case, the integer optimizer terminates and reports the integer
feasible solution as an optimal solution. Please note that z is a valid upper bound, and
z a valid lower bound, to the optimal objective value z

. In minimization problems, z
normally increases gradually during the solution process, while z decreases each time a
better integer solution is located starting at the warm-started or heuristically preprocessor
generated solution value.
The tolerances are specied using parameters and have default values as shown below.
If an optimal solution can not be located within a reasonable amount of time, it may be
advantageous to employ a relaxed termination criterion after some time. Whenever the
integer optimizer locates an integer feasible solution, and has spent at least the number
of seconds dened by the double-typed parameter MIO_DISABLE_TERM_TIME on
solving the problem, it will check whether the criterion
Tolerance Parameter name (type: dparam) Default value

1
MIO_TOL_ABS_RELAX_INT 10
5

2
MIO_TOL_REL_RELAX_INT 10
6

3
MIO_TOL_ABS_GAP 0

4
MIO_TOL_REL_GAP 10
4

5
MIO_NEAR_TOL_ABS_GAP 0

6
MIO_NEAR_TOL_REL_GAP 10
3
Table 2.20: Integer optimizer tolerances.
Default values are subject to change from MOSEK v7.0 shown here.
47
2.5 Mixed Integer Programming A GUIDED TOUR
[z z[ max (
5
,
6
max (1, [z[)) (2.21)
is satised. If it is satised, the optimizer will report that the candidate solution is near
optimal and then terminate. Please note that the relaxed termination criterion is not
active by default, as the delay (MIO_DISABLE_TERM_TIME) is -1 by default.
Given the delay (MIO_DISABLE_TERM_TIME) is dierent from -1, the integer
optimizer also oers two additional parameters to stop the optimizer prematurely once the
specied delay have passed. Beware that these parameters, in the table below, are not able
to guarantee any kind of near optimality, and with a default value of -1 they are ignored
until explicitly dened.
Parameter name (type: iparam) Maximum number of ...
MIO_MAX_NUM_BRANCHES Branches allowed.
MIO_MAX_NUM_RELAX Relaxations allowed.
Another option is to terminate whenever the number of distinct integer feasible solution
reaches the value of the parameter MIO_MAX_NUM_SOLUTIONS. Note that this
criteria does not depend on the delay (MIO_DISABLE_TERM_TIME).
Parameter name (type: iparam) Maximum number of ...
MIO_MAX_NUM_SOLUTIONS Integer feasible solutions allowed.
Finally, if all other termination criteria fail to be satised, the double-typed parameter
OPTIMIZER_MAX_TIME can be used to dene the maximum number of seconds to
spend before terminating with the best integer feasible solution (if any). See Section 2.6
on how to specify all these parameters.
48
2.6 Parameter settings in MOSEK A GUIDED TOUR
2.6 Parameter settings in MOSEK
The MOSEK optimization library oers a lot of customization for the user to be able to
control the behavior of the optimization process or the returned output information. All
parameters have been documented on the homepage of MOSEK
20
, and are all supported
by this interface. Only a few is mentioned here.
Notice that the MSK_ prex, required by the MOSEK C API, can be ignored
for string-typed parameter values in this interface. Similarly the MSK_IPAR_,
MSK_DPAR_ and MSK_SPAR_ prexes for parameter names, have been removed
in favor of the iparam, dparam and sparam structures. Also in this interface, parameters
are case-insensitive. This means that parameter names and string-typed parameter values
can be written in both upper and lower case.
The LOG-Parameter Example This is a logging parameter controlling the amount
of information printed to the screen from all channels within the MOSEK optimization
library. The value of the parameter can be set to any integer between 0 (suppressing all
information) to the R value Inf (releasing all information), and the default is 10.
Revisiting the lo1 example from Section 2.2.1, we can now try to silence the
optimization process as shown below.
Figure 24: Suppressing the optimization log with parameter LOG
lo1 <- list(sense = "max")
lo1$c <- c(3,1,5,1)
lo1$A <- Matrix(c(3,1,2,0,
2,1,3,1,
0,2,0,3), nrow=3, byrow=TRUE , sparse=TRUE)
blc <- c(30,15,-Inf)
buc <- c(30,Inf ,25); lo1$bc <- rbind(blc , buc);
blx <- c(30,15,-Inf)
bux <- c(30,Inf ,25); lo1$bx <- rbind(blx , bux);
lo1$iparam <- list(LOG = 0);
r <- mosek(lo1);
Notice that errors and warnings from the interface itself will not be aected by this
parameter. These are controlled separately by the option verbose.

20
Check out: mosek.com Documentation C API manual Parameter reference.
49
2.6 Parameter settings in MOSEK A GUIDED TOUR
Adding parameters to the input arguments of a convex optimization problem is, as seen,
straightforward (refer to Figure 2). The list iparam should specify its variables with a
name, matching that of an integer-typed parameter, and a value, being the corresponding
parameter value. Similarly the list dparam is for double-typed parameters, and sparam
is for string-typed parameters. Notice that reference-strings to an enum (a small set of
named possibilities) is also integer-typed, as the input reference-strings are automatically
converted to indexes by the interface.
The OPTIMIZER-Parameter Example This parameter controls which optimizer
used to solve a specic problem. The default value is OPTIMIZER_FREE meaning that
MOSEK will try to gure out what optimizer to use on its own.
In this example we shall try to solve the lo1 example from Section 2.2.1 again, only this
time using the dual simplex method. This is specied by setting the parameter to the
enum reference-string value OPTIMIZER_DUAL_SIMPLEX as shown.
Figure 25: Selecting the dual simplex method with parameter OPTIMIZER
lo1 <- list(sense = "max")
lo1$c <- c(3,1,5,1)
lo1$A <- Matrix(c(3,1,2,0,
2,1,3,1,
0,2,0,3), nrow=3, byrow=TRUE , sparse=TRUE)
blc <- c(30,15,-Inf)
buc <- c(30,Inf ,25); lo1$bc <- rbind(blc , buc);
blx <- c(30,15,-Inf)
bux <- c(30,Inf ,25); lo1$bx <- rbind(blx , bux);
lo1$iparam <- list(OPTIMIZER = "OPTIMIZER_DUAL_SIMPLEX ");
r <- mosek(lo1)

Setting a parameter to NULL will remove it from the list according to the R language
denition. Setting a parameter to NA or NaN, will on the other hand keep it on the list,
only to be ignored by the interface with warnings conrming that this took place. Errors
will be generated when a parameter name is not recognized or when the value dened for
it is not within its feasible range.
50
2.7 Exporting and Importing optimization models A GUIDED TOUR
2.7 Exporting and Importing optimization models
This section concerns the export and import of optimization models in all standard
modeling formats (e.g. lp, mps and opf) and the MOSEK Task format task. These
les can be converted to and from the problem descriptions (R-variables) that the mosek
function can accept (refer to Figure 2).
lp mps opf task
Linear problem
Conic constraint
Semidenite variable
Integer variable
MOSEK Parameter
Solution
Figure 26: Supported le formats recognized by their le extension.
As seen in Figure 26, some of these modeling formats have more limitations than others.
The lp format only support linear problems and integer variables. The mps format is
more general with support of conic constraints and vendor-specic parameter settings. The
opf format further allow solutions to be dened, but only the task format handles the
full problem description with semidenite variables.
In all formats supporting parameter settings, either none or the entire list of all
parameters in MOSEK are exported. This will lead to large problem descriptions when
parameters are imported from a le. To save and reload the constructed R-variables exactly
as they are, please use the save and load commands part of the R language denition.
2.7.1 The write and read functionality
Exporting through the mosek_write function
A problem description can be exported with the mosek_write command taking the problem
variable and destination as input, and returning a response that can be used to verify
success. The destination should be a lepath with a le extension matching the modeling
format to be used. If the le extension is not recognized as one of the supported modeling
formats, the mps format is used by default. Note, however, that this only aects the
contents of the le and does not change the le extension to mps. The specied lepath
is overwritten if it already exists, and otherwise created.
51
2.7 Exporting and Importing optimization models A GUIDED TOUR
Exporting the lo1 Example Revisiting the lo1 example from Section 2.2.1, we can
now try to export the optimization model to the opf format. In the current example we
export the problem description to a le called lo1.opf in the current working directory.
Beware that this le is overwritten if it already exists! Afterwards, using the standard
file.show command, the le is displayed in the R-console.
Figure 27: Exporting a Linear Optimization Problem
lo1 <- list(sense = "max")
lo1$c <- c(3,1,5,1)
lo1$A <- Matrix(c(3,1,2,0,
2,1,3,1,
0,2,0,3), nrow=3, byrow=TRUE , sparse=TRUE)
blc <- c(30,15,-Inf)
buc <- c(30,Inf ,25); lo1$bc <- rbind(blc , buc);
blx <- c(0,0,0,0)
bux <- c(Inf ,10,Inf ,Inf); lo1$bx <- rbind(blx , bux);
r <- mosek_write(lo1 , "lo1.opf")
file.show("lo1.opf")

In the example we did not have either parameters or initial solutions dened in the problem
description, but if we had, the options usesol and useparam could be used to indicate
whether to write them to the le or not. By default, usesol is true and useparam is
false. Parameters are thus usually omitted in exported models. Note that since it does
not make sense for an initial solution to specify a solution or problem status, the solutions
exported as part of a problem description will always have unknown statuses. The export
functionality in the mosek function behaves dierently in this regard (see Section 2.7.2) .
Importing through the mosek_read function
A problem description can be imported with the mosek_read command taking the lepath
as input. If the le extension does not match one of the supported modeling formats, the
mps format is assumed and errors are given if the contents of the le does not respect
this syntax.
52
2.7 Exporting and Importing optimization models A GUIDED TOUR
Importing the lo1 Example (Part 1 of 2) We will now try to import the lo1
optimization model that was exported in Figure 27. In the current example we import
from a le called lo1.opf in the current working directory. Notice that we use the options
usesol and useparam to indicate that we do not wish to read solutions from the le, but
are interested in the parameter settings. Checking that the response code is ne, we print
a message and assign the lo1 variable. Please note the size of the elds iparam, dparam
and sparam of the imported problem description.
Figure 28: Importing a Linear Optimization Problem
r <- mosek_read ("lo1.opf", list(usesol=FALSE , useparam=TRUE))
if (identical(r$response$code , 0)) {
print (" Successfully read the optimization model ")
lo1 <- r$prob
}

The options usesol and useparam behaves as for the mosek_write function, and can be
used to specify whether to read the solutions and parameter settings from the le. By
default, usesol is true and useparam is false. Note that for all parameters not dened in
the le, the default value in the MOSEK optimization library is assigned instead. This
means that the returned problem description will always be quite large when parameters
are imported.
Another option of special interest is the string-typed matrixformat which can specify the
format of the imported sparse constraint matrix called A in Figure 2. The recognized values
for this option are shown below with pkgMatrix:COO (abbreviated COO) being the default.
All string values are case insensitive.
COO / pkgMatrix:COO
The sparse coordinate format (aka triplet format) from package Matrix.
CSC / pkgMatrix:CSC
The compressed sparse column format from package Matrix.
simple:COO
The sparse coordinate format (aka triplet format) based on list-objects.
When setting matrixformat to the string-value simple:COO, the constraint matrix will
be a named list with scalars nrow and ncol telling respective the number of rows and
columns of matrix A, and lists i, j and v telling the ij-coordinates of each value in v. All
these matrix formats are allowed in the problem description given as input to the mosek
and mosek_write functions.
53
2.7 Exporting and Importing optimization models A GUIDED TOUR
Importing the lo1 Example (Part 2 of 2) We will again try to import the lo1
optimization model that was exported in Figure 27, only this time ignoring all parameters
and with the matrixformat option set to simple:COO. This is shown in Figure 29 where
a le called lo1.opf is imported from the current working directory. The structure of the
imported constraint matrix is shown in Figure 30.
Figure 29: Importing a Linear Optimization Problem with simple:COO
r <- mosek_read ("lo1.opf", list(matrixformat =" simple:COO "))
Figure 30: Importing a Linear Optimization Problem with simple:COO
r$prob$A
{
$nrow
[1] 3
$ncol
[1] 4
$i
[1] 1 2 1 2 3 1 2 2 3
$j
[1] 1 1 2 2 2 3 3 4 4
$v
[1] 3 2 1 1 2 2 3 1 3
}

2.7.2 Writing immediately before/after optimizing

As an alternative to mosek_write, the mosek function itself can be used to export the
interpreted optimization model and identied solution to in all supported modeling formats
(e.g. lp, opf, mps and task). Note that the here mentioned options only apply to the
mosek function.
The option writebefore will export the model exactly as MOSEK have received it,
immediately before it starts the optimization procedure. This can for instance be used
to inspect the problem description in formats that the user may be more familiar with,
to see if what was intended, matched what had been expressed. This option can also be
used as a log-le of the last solved model, such that if the MOSEK optimization library
somehow behaves unexpected, the input model can easily be reported.
The option writeafter will export the model and identied solutions, immediately after
the optimization procedure has ended. In contrast to the mosek_write function, these
54
2.7 Exporting and Importing optimization models A GUIDED TOUR
solutions are not seen as initial solutions, and do have the correct solution and problems
statuses exported with them. Notice that only the opf and task formats support solutions
(Figure 26), and that all other modeling formats would thus result in les similar to those
written by writebefore.
Finally, it may be of interest, that the options usesol and useparam can also be used in this
context. These options were explained in the description of mosek_write and mosek_read
previously in this chapter.
55
2.8 Frequently Asked Questions A GUIDED TOUR
2.8 Frequently Asked Questions
2.8.1 How to make Rmosek silent?
The verbosity of the interface - that is, how many details that are printed - can be regulated
by the option verbose (see e.g. Figure 2, Section 2.1). The default value is 10, but as of now
there only exists four message priorities: Errors=1, MOSEK=2, Warnings=3 and Info=4.
Setting verbose to the value of 1, will for example only allow the interface to speak when
errors are encountered. Setting verbose to 2, will also open for the output stream from
the MOSEK optimization library. Do note that verbosity levels less than three will ignore
warnings which may hold valuable information.
NB! Setting the option verbose = 0 will completely silence everything...
(even when errors and warnings have occurred)
The verbosity of the MOSEK optimization library, can be regulated by a large range of
parameters
21
. In many cases the integer-typed LOG-parameter provides sucient control
over the logging details and can be set as shown in Figure 24, Section 2.6.
2.8.2 How to force a premature termination?
While the MOSEK optimization library is running, the interface will once in a while check
for the keyboard sequence <CTRL> + <C> used by the user to signal that a premature
exit would be preferable. Once the signal is registered the error message Interruption
caught, terminating at rst chance... will be printed to the screen and the best solution
found so far will be processed and returned.
2.8.3 How to change the termination criteria?
The list of termination criteria is long, spanning over absolute and relative tolerances,
maximum iterations, time limits and objective bounds
22
. Each criterion can be modied
through a parameter (see Section 2.6 for help on this). As an example, the integer-typed
parameters SIM_MAX_ITERATIONS and INTPNT_MAX_ITERATIONS respectively
controls the maximum numbers of simplex and interior-point iterations, while the double-
typed parameter OPTIMIZER_MAX_TIME sets the overall time limit in seconds. For
mixed integer optimization problems, more information can be found in Section 2.5.3.
21
Check out: mosek.com Documentation C API manual Logging parameters.
22
Check out: mosek.com Documentation C API manual Termination criterion parameters.
56
2.8 Frequently Asked Questions A GUIDED TOUR
2.8.4 How to nd the objective value and other information items?
Extra details from a call to the mosek function, can be requested using the options soldetail
and getinfo (see e.g. Figure 2, Section 2.1).
Currently there exists three levels of solution detail: NoExtras=0 (the default),
ObjectiveValues=1 and MaxInfeas=2. Setting soldetail to at least 1, will add the primal
objective value pobjval and the dual objective value dobjval (or another primal objective
bound pobjbound in case of mixed-integer optimization.). Setting soldetail to at least 2,
furthermore adds the list of maximal infeasibilities maxinfeas. This list covers the primal
inequality, equality and conic constraints (pbound, peq and pcone), the dual inequality,
equality and conic constraints (dbound, deq and dcone), as well as the fractionality of
primal integer variables int.
Setting getinfo to the boolean value TRUE, will add two lists with information items
extracted from the MOSEK optimization library. One contains all available integer-typed
information iinfo, based on items with a prex of MSK_IINF_ and MSK_LIINF_ in the
MOSEK C API
23
. The other list contains all available double-typed information dinfo,
based on items with a prex of MSK_DINF_ in the MOSEK C API.
2.8.5 How to report a bug or get further help?
We invite all users of the R-to-MOSEK Interface to join the MOSEK Google group
at http://groups.google.com/group/mosek. The use of the Mosek Google group is
the preferred form of contact, particularly for academics, as it reaches a broader user
community which may be able to help or provide feedback to your posts. Otherwise,
MOSEK ApS can be reached at support@mosek.com.
23
Check out: mosek.com Documentation C API manual API constants.
57
ADVANCED TOPICS
3 Advanced topics
3.1 Parallel optimization
3.1.1 The multicore package
The R package called multicore, provides functions for parallel execution of R code on
machines with multiple cores or CPUs. Windows is not currently supported, but the
package should work on most UNIX-alike platforms.
The multicore package works by copying the full memory state of the R session to
new processes. While this seems like a large overhead, in practice, the copy is delayed
until modication assuring a smooth parallel execution. The downside is that this low-
level memory state copy is not safe for all types of resources. As an example, parallel
interactions with the GUI or on-screen devices can cause the R session to crash. It is thus
recommended only to use the multicore package in console R.
In the Rmosek package a license is an externally acquired resource, and attempts to
simply copy the memory state of this resource will provoke a session crash. Thus, licenses
should always be released before the time of parallelization.
NB! Always call mosek_clean() before a parallelizing operator.
(failure to do so is likely to provoke session crashes)
A consequence of this is that each new process will be using a separate license. That is,
your license system should allow 8 licenses to be checked out simultaneously, if you wish
to solve 8 optimization problems in parallel. Please note that unlimited academic and
commercial licenses are available at MOSEK.
Solving the lo1 Example in Parallel (Part 1 of 2) The following is an example of
a linear optimization problem with one equality and two inequality constraints:
maximize 3x
1
+ 1x
2
+ 5x
3
+ 1x
4
subject to 3x
1
+ 1x
2
+ 2x
3
= ,
2x
1
+ 1x
2
+ 3x
3
+ 1x
4
15,
2x
2
+ 3x
4
25,
(3.1)
having the bounds
0 x
1
,
0 x
2
10,
0 x
3
,
0 x
4
.
(3.2)
58
3.1 Parallel optimization ADVANCED TOPICS
This problem is infeasible for < 40/9 4.44. As shown in Figure 31, the problem is
solved by 4 parallel processes for 10 dierent values of stepping from 3 to 30.
Figure 31: Parallel Optimization (permutations of lo1)
require(Rmosek)
lo1 <- list(sense = "max")
lo1$c <- c(3,1,5,1)
lo1$A <- Matrix(c(3,1,2,0,
2,1,3,1,
0,2,0,3), nrow=3, byrow=TRUE , sparse=TRUE)
blc <- c(30,15,-Inf)
buc <- c(30,Inf ,25); lo1$bc <- rbind(blc , buc)
blx <- c(0,0,0,0)
bux <- c(Inf ,10,Inf ,Inf); lo1$bx <- rbind(blx , bux)
#
# Solve permutations of this problem in parallel
#
require(foreach ); require(doMC)
registerDoMC(cores =4)
mosek_clean ()
r <- foreach(lambda = seq(3,30, length =10)) %dopar%
{
lo1$bc [,1] <- c(lambda , lambda)
return( mosek(lo1) )
}
Note that we use a loop construct from the foreach package, with the package doMC
(depending on package multicore) registered as the parallel back-end. This is easier than
using the multicore package directly. Please see the documentation of the foreach package
for more details on how to use this.
Furthermore, note the mosek_clean() call immediately before the %dopar% operator which
is highly recommended for a smooth parallel execution.

Controlling the output stream

One of the issues with parallel programming is that if multiple processes use the same
output stream, the messages will be mixed up and hard to tell apart. One solution is to
use the capture.output function part of the standard R installation, and write the output
stream of each thread to a separate le.
59
3.1 Parallel optimization ADVANCED TOPICS
Solving the lo1 Example in Parallel (Part 2 of 2) In this example we use
the capture.output function to write the output of each parallel process to les called
RESULTS_FILE_ for the specic in use. These les will placed in the current working
directory of the R session.
Figure 32: Parallel Optimization (permutations of lo1)
# Define problem and register parallel back -end as before ..
capture.output.helper <- function(file , code) {
capture.output(file=file , res <- code(), mosek_clean (), print("Done "))
return(res)
}
mosek_clean ()
r <- foreach(lambda = seq(3,30, length =10)) %dopar% capture.output.helper(
file = paste(" RESULTS_FILE_", lambda , sep=""),
code = function ()
{
lo1$bc [,1] <- c(lambda , lambda)
return( mosek(lo1) )
}
)
The dened capture.output.helper solves two problems. It makes sure that the
return value is correctly propagated to the foreach-loop, as opposed to the standard
capture.output function ignoring return values. It also calls mosek_clean(), to capture
any cleaning messages from Rmosek, which would otherwise appear on screen when the
parallel process is terminated outside the control of capture.output.

60
3.2 Quadratic Convex Optimization ADVANCED TOPICS
3.2 Quadratic Convex Optimization
3.2.1 Solving QP problems
Quadratic Convex Optimization is a generalization of linear programming for which
quadratic terms can be added to the objective function. Convex here emphasize that
the entire formulation should dene a convex optimization problem.
minimize 0.5x
T
Qx + c
T
x + c
0
subject to l
c
Ax u
c
,
l
x
x u
x
.
(3.3)
Two facts should be noted, before engaging in quadratic convex optimization:
The problem is convex only if Q is positive semidenite (all eigenvalues non-negative),
which can be numerically hard to verify for large Q with zero or near-zero eigenvalues.
The MOSEK optimization library does not support the combination of quadratic
convex optimization with conic constraints.
If the problem at hand involves quadratic objective terms, quadratic convex optimization
may very well be the best way to solve it. Nevertheless, it is easy to try the conic quadratic
formulation of these problems, for which the optimizer will sometimes perform even better
(see Section 2.3.3).
The qo1 Example The following is an example of a linear constrained problem with
a convex quadratic objective.
minimize 0.5
_
2x
2
1
2x
3
x
1
+ 0.2x
2
2
+ 2x
2
3
_
x
2
subject to x
1
+ x
2
+ x
3
1 ,
x
1
, x
2
, x
3
0 .
(3.4)
The quadratic objective terms are easily appended on top of the linear formulation as
shown in the following R code (Figure 33).
61
3.2 Quadratic Convex Optimization ADVANCED TOPICS
Figure 33: Quadratic Convex Optimization (qo1)
qo1 <- list()
qo1$sense <- "min"
qo1$c <- c(0,-1,0)
qo1$A <- Matrix(c(1,1,1), nrow=1, byrow=TRUE , sparse=TRUE)
qo1$bc <- rbind(blc = 1,
buc = Inf)
qo1$bx <- rbind(blx = rep(0,3),
bux = rep(Inf ,3))
qo1$qobj <- list(i = c(1, 3, 2, 3),
j = c(1, 1, 2, 3),
v = c(2, -1, 0.2, 2))
r <- mosek(qo1)
The objective term 2x
3
x
1
has been specied with half the coecient, i.e. 1, in the eld
qobj. Note that this is in contrary to squared variables, e.g. 2x
2
1
, as explained in the text.

From this example the input arguments for a quadratic convex program (3.3) follow easily
(refer to Figure 2, Section 2.1). The linear part of the objective function and constraints,
as well as the constraint and variable bounds, should all be specied as for linear programs
(see Section 2.2). The only addition to this is the list called qobj containing the triplets
of row indexes i, column indexes j and values v that dene Q. The number of rows and
columns of matrix Q should not be specied, as these dimensions are already known from
the number of variables in the linear part of the problem.
The inputted Q is always assumed symmetric, and note that this is without loss of
generality. This is ecient as only the lower triangular part of Q thus needs to be specied.
Setting a coecient at the lower triangular position (i,j) will, by symmetry, imply that the
same coecient is also present at position (j,i). In order to add the objective term 2x
3
x
1
,
one would see it as x
3
x
1
x
1
x
3
and only input a coecient of 1 to row 3 of column 1.
62
3.3 Separable Convex Optimization ADVANCED TOPICS
3.3 Separable Convex Optimization
3.3.1 Solving SCOPT problems
Separable Convex Optimization is a generalization of linear programming for which
arbitrary unary operators can be added to the objective function and constraints. Separable
means that each operator should depend on one variable only, and Convex means that the
entire formulation should dene a convex optimization problem.
minimize z(x) + c
T
x + c
0
subject to l
c
w(x) + Ax u
c
,
l
x
x u
x
,
(3.5)
where
z(x) =
n

j=1
z
j
(x
j
) , z
j
: R R.
w(x) =
_
_
_
.
.
.
w
i
(x)
.
.
.
_
_
_ , w
i
(x) =
n

j=1
w
ij
(x
j
) , w
ij
: R R.
(3.6)
Three facts should be noted, before engaging in separable convex optimization:
Since the operators z
j
(x
j
) and w
ij
(x
j
) are arbitrarily dened, the optimizer for
separable convex optimization has less knowledge, and can be expected to perform
worse, than the optimizer for conic quadratic programs.
No standard modeling le-format exists for separable convex optimization.
(MOSEK has its own sco format dened for a subset of operators)
The MOSEK optimization library does not support the combination of separable
convex optimization with integer variables or conic constraints.
If the problem at hand involves exponential or logarithmic functions, separable convex
optimization may be a good and ecient choice. Note, however, that undened behaviour
can be observed if the problem is not convex (MOSEK will try to detect this). If the
problem does not involve involves exponential or logarithmic functions, it may very well
be possible to reformulate the optimization problem as a conic quadratic program (see e.g.
Section 2.3 and Appendix B).
63
3.3 Separable Convex Optimization ADVANCED TOPICS
For the sake of performance, and the usability of the sco le-format, callback functionality
to custom R evaluation code have not been supported. Instead, a list of common operators
with customizable coecients have been added to this interface, making it both fast and
easy to use. The reader is encouraged to contact us, in case this list of operators is found
to be insucient.
With constants f R, g R and h R, the interface currently allows:
Operator Denition Domain
Entropy fxln(x) 0 < x <
Exponential f exp(gx + h) < x <
Logarithm f ln(gx + h) If g > 0: h/g < x <
Otherwise: < x < h/g
Power f(x + h)
g
If g > 0 and integer: < x <
If g < 0 and integer: choose this h < x <
or this < x < h
Otherwise: h < x <
Figure 27: Supported unary operators.
The sco1 Example The following is an example of a problem with a non-linear
objective optimized over a linear constraint and the interior of a unit circle.
minimize x
1
ln(x
3
)
subject to x
2
1
+ x
2
2
1 ,
x
1
+ 2x
2
x
3
= 0 ,
x
3
0 .
(3.7)
The objective contains one logarithm operator, and the rst constraint contains two power
operators. These operators are easily appended on top of the linear formulation as shown
in the following R code (Figure 34).
64
3.3 Separable Convex Optimization ADVANCED TOPICS
Figure 34: Separable Convex Optimization (sco1)
sco1 <- list(sense = "min")
sco1$c <- c(1,0,0)
sco1$A <- Matrix(c(0, 0, 0,
1, 2, -1), nrow=2, byrow=TRUE , sparse=TRUE)
sco1$bc <- rbind(blc = c(-Inf ,0),
buc = c(1,0))
sco1$bx <- rbind(blx = c(-Inf ,-Inf ,0),
bux = rep(Inf ,3))
NUMOPRO <- 1; NUMOPRC <- 2;
opro <- matrix(list(), nrow=5, ncol=NUMOPRO)
oprc <- matrix(list(), nrow=6, ncol=NUMOPRC)
rownames(opro) <- c("type","j","f","g","h")
rownames(oprc) <- c("type","i","j","f","g","h")
opro[,1] <- list("LOG", 3, -1.0, 1.0, 0.0)
oprc[,1] <- list("POW", 1, 1, 1.0, 2.0, 0.0)
oprc[,2] <- list("POW", 1, 2, 1.0, 2.0, 0.0)
sco1$scopt <- list(opro=opro , oprc=oprc)
r <- mosek(sco1)

From this example the input arguments for a separable convex program (3.5) follow easily
(refer to Figure 2, Section 2.1). The linear part of the objective function and constraints,
as well as the constraint and variable bounds, should all be specied as for linear programs
(see Section 2.2). The only addition to this is the list called scopt containing the list-typed
operator matrices opro (for objective) and oprc (for constraints).
The operator matrices have a column for each operator and a row for each descriptive
element. The opro matrix have ve rows called type, j, f, g, h, while the oprc matrix have
six rows called type, i, j, f, g, h. Row type should specify the operator type in a string,
being either entropy "ENT", exponential "EXP", logarithm "LOG" or power "POW". Row
i (not in opro) should specify the index of the constraint to which the non-linear term
should be added. Row j should specify the variable index of the operator. Rows f , g and
h should specify the coecients of the operator (see Figure 27).
Examples are:
Terms in opro type j f g h
In objective: exp(1.4x
1
+ 1.1) "EXP" 1 1.0 1.4 1.1
In objective: ln(1.5x
2
+ 1.2) "LOG" 2 1.0 1.5 1.2
In objective: 2.1

x
3
"POW" 3 2.1 0.5 0.0
65
3.3 Separable Convex Optimization ADVANCED TOPICS
Terms in oprc type i j f g h
In constraint one: 0.1x
1
ln(x
1
) "ENT" 1 1 0.1 NA NA
In constraint two: (x
1
+ 2.0)
1.75
"POW" 2 1 1.0 1.75 2.0
Note that the denition of the entropy operator (see Figure 27), was the only operator
dened without g and h. Thus, for entropy operators, these two unused rows in the operator
matrix can be set to either zero or any empty denition (NULL, NA or NaN).
3.3.2 Convexity, dierentiability and safe bounds
Ensuring convexity
The feasible solution space of a separable convex optimization problem is required to be
convex. If a simple set of convexity-preserving rules are complied with, the problem may
further be called disciplined convex. Disciplined convexity is not required in this interface,
but will be checked to help the user identify modeling issues.
Disciplined convexity means that the expression g
i
(x) of a constraint l
c
i
g
i
(x) u
c
i
, is
required to be concave if the lower bound is nite (i.e. not -Inf), and required to be convex
if the upper bound is nite (i.e. not Inf). This also implies that the expression should be
linear if both the lower and upper bound is dened with nite values.
In case of separable convex optimization, this means that each unary operator of an
expression - required to be convex (resp. concave) - is used only in its convex (resp.
concave) domain. A problem which is disciplined convex is guaranteed to be convex.
Operator Denition Convexity (domains dened in Figure 27)
Entropy fxln(x) Convex in domain.
Exponential f exp(gx + h) Convex in domain.
Logarithm f ln(gx + h) Concave in domain.
Power f(x + h)
g
If g is even integer: convex in domain.
If g is odd integer: concave (x < h), convex (h < x)
If 0 < g < 1: concave in domain.
Otherwise: convex in domain.
Figure 28: Convexity of unary operators assuming f > 0.
Convexities are exactly opposite for f < 0.
66
3.3 Separable Convex Optimization ADVANCED TOPICS
Ensuring dierentiability with safe bounds
Figure 27 displayed the domains in which each operator was well-dened and dierentiable.
If the MOSEK optimizer for some reason tried to evaluate one of these operators outside
of its domain, an error would occur and the optimization would terminate. Since MOSEK
may not be able to guess an initial feasible solution to the problem, it may actually have
to work its way through the infeasible region of the problem.
Essentially, operators may be evaluated in points where the constraint bounds are
not satised. To avoid causing the optimizer to terminate due to evaluations outside
operator domains, variable bounds should be specied such that operator domains are
always satised. This is called safe bounds, and will work because the MOSEK optimizer
is guaranteed to satisfy variable bounds in each iteration.
3.3.3 Importing and exporting modelles
In case the problem description contains separable convex operators, the string-typed
option scofile must be used to specify the lepath to which these operators should be
written. Failing to specify this option when calling mosek_write (see Section 2.7) will
result in an error. It is recommended to use the sco le extension, but the interface will
allow any (even a missing one) to be used without warning.
Exporting the sco1 Example Revisiting the sco1 example from Section 3.3.1, we
can now try to export the optimization model to the opf format with operators in an sco
le. In particular, the current example will export the problem description to respectively
sco1.opf and sco1.sco in the current working directory. Beware that these les are
overwritten if they already exists! Afterwards, using the standard file.show command,
the les are displayed in the R-console.
Figure 35: Exporting a Separable Convex Optimization Problem
# Define sco1 (not shown)
r <- mosek_write(sco1 , "sco1.opf", list(scofile ="sco1.sco"))
file.show("sco1.opf", "sco1.sco")

The same option, scole, can be used when calling mosek_read (see Section 2.7). This
function will read in a problem description of a separable convex optimization problem from
the location of two les. These two les can also be read independently of each other.
67
DEVELOPERS CORNER
4 Developers corner
4.1 Introduction
In the projects src folder you can nd the header and source les (*.h and *.cc) containing
the C++ code of the R-to-MOSEK interface. Moreover, the Rmosek.cc le contains the
functions that are callable from R.
This code has been written and commented so that it should be easy to understand
how the interface handles the input and output structures, prints to the screen and
communicates with the MOSEK optimization library. Four important references proved
to be useful in the development of this interface.
The MOSEK C API Reference
24
R Language Denition
25
Writing R Extensions
25
R Installation and Administration
25
The program is free software and has been released under the terms of the GNU Lesser
General Public License as published by the Free Software Foundation, either version 2.1 of
the License, or any later version. You are welcome to extend upon this release and modify
the interface to your personal needs. The source code is distributed as part of the Rmosek
package.
Notes on the Matrix package: A number of bugxes were introduced in the Matrix
package posterior to dependencies on R(>=2.12.0). These xes have been backported in
this package, to older versions of the package Matrix(>=0.9996875-3) depending only on
R(>=2.10.0). Eectively, more users will be able to run Rmosek. The source code for this
backport is located in the compatibility folder from which it is included in the local_stubs.h
and local_stubs.cc les.
24
Check out: mosek.com Documentation C API manual API reference.
25
Check out: r-project.org Manuals
68
4.2 Limitations and Missing Features DEVELOPERS CORNER
4.2 Limitations and Missing Features
The interface currently provides the features needed for basic research and educational pur-
poses, but has many shortcomings when compared to the Matlab-to-MOSEK interface
26
.
This is a short list of missing features.
1. Direct input of Quadratic Constrained Programs: At the moment the user will
need to manually transform the Quadratic Constrained Problem to a Conic Program
as the interface only provides for the direct input of linear and conic programs.
The MOSEK optimization library actually supports the direct input of Quadratic
Constrained programs, so this should only require a small amount of coding.
2. Custom R-coded callback functions: While the MOSEK optimization library
executes an optimization procedure, it is possible to dene a callback function to
check on the progress and terminate ahead of time. It could be useful to make this
possibility available from R, such that a callback option pointing to an R-function
could be specied.
26
Check out: mosek.com Optimization toolbox for MATLAB.
69
COMMAND REFERENCE
A Command reference
r <- mosek_version()
Retrieves a string containing the version number of the utilized MOSEK optimization
library. Briey mentioned in Section 2.1.
r <- mosek(problem, opts = list())
Solve an optimization problem using the MOSEK optimization library. The input variable
problem could have any name, but should be a list object describing the optimization
problem using the following elds.
problem Problem description
..$sense Objective sense, e.g. "max" or "min"
..$c Objective coecients
..$c0 Objective constant
..$A Constraint matrix
..$bc Lower and upper constraint bounds
..$bx Lower and upper variable bounds
..$cones Conic constraints
..$bardim Semidenite variable dimensions
..$barc Semidenite objective coecients
..$barA Semidenite constraint coecients
..$intsub Integer variable indexes
..$qobj Quadratic convex optimization
..$scopt Separable convex optimization
..$iparam/$dparam/$sparam Parameter list
....$<MSK_PARAM> Value of any <MSK_PARAM>
..$sol Initial solution list
....$itr/$bas/$int Initial solution description
The argument sense is the goal of the optimization and should indicate whether we wish
to maximize or minimize the objective function given by f(x) = c
T
x + c
0
, where c is the
objective coecients and c0 the objective constant. The matrix A together with bounds
bc describes the linear constraints of the problem, while variable bounds are given by bx.
These input arguments describe a linear program (see Section 2.2).
The argument cones is used for Conic Programming and has been described in Section
2.3. The issue regarding the transformation of other convex problems to the formulation of a
Conic Program has been addressed in Appendix B, with more details on the transformation
of Quadratic Programs in Section 2.3.3.
70
COMMAND REFERENCE
The arguments bardim, barc and barA are used for Semidenite Programming and
have been described in Section 2.4.
The argument intsub is used to specify whether some of the variables should only be
allowed to take integer values as part of a feasible solution. This is necessary for the class
of Mixed Integer Programs described in Section 2.5.
The argument qobj is used for quadratic convex optimization and allows quadratic
terms in the objective function. This argument has been described in Section 3.2.
The argument scopt is used for separable convex optimization and allows non-linear
unary operators in formulations involving exponential or logarithmic terms. This argument
has been described in Section 3.3.
The arguments iparam, dparam and sparam are used to specify lists of integer-,
double- and string-typed parameters for the MOSEK optimization library in order to aid
or take control of the optimization process. This has been described in Section 2.6.
The argument sol is used to specify an initial solution used to hot-start the optimization
process, which is likely to increase the solution speed. This has been described for linear
programming in Section 2.2.4 and for mixed integer programming in Section 2.5.2.
The options opts could have any name, and are, in fact, often input directly as an
anonymous list. It has the following elds.
opts Options for the mosek function
..$verbose Output logging verbosity
..$usesol Whether to use the initial solution
..$useparam Whether to use the specied parameter settings
..$soldetail Level of detail used to describe solutions
..$getinfo Whether to extract MOSEK information items
..$writebefore Filepath used to export model
..$writeafter Filepath used to export model and solution
The argument verbose is used to specify the amount of logging information given by
the interface. This is described in Section 2.8.1.
The argument usesol can be used to ignore the problem eld sol, while useparam
can be used to ignore the elds iparam, dparam and sparam, without having to change the
problem description. This usage has not been covered by the user guide, but it should be
noted that both options are true by default in contrary to the context of Section 2.7.
The arguments soldetail and getinfo are used to control the amount of information
returned about the solution, and the MOSEK optimization library internals, respectively.
This is described in Section 2.8.4.
71
COMMAND REFERENCE
The arguments writebefore and writeafter are used to see the optimization model
(and identied solution) constructed by MOSEK, written out to a le just before (or
immediately after) the optimization process. This is described in Section 2.7.
The resulting function output variable r, returned by the interface, is capable of holding
the three types of solutions listed below, along with the response of the function call. The
existence of a solution depends on the optimization problem and the algorithm used to
solve it.
r Result of the mosek function
..$response Response from the MOSEK optimization library
....$code ID-code of response
....$msg Human-readable message
..$sol Solution list
....$itr/$bas/$int Solution description
......$solsta Solution status
......$prosta Problem status
......$skc Linear constraint status keys
......$skx Variable bound status keys
......$skn Conic constraint status keys
......$xc Constraint activities
......$xx Variable activities
......$barx Semidenite variable activities
......$slc Dual variable for constraint lower bounds
......$suc Dual variable for constraint upper bounds
......$slx Dual variable for variable lower bounds
......$sux Dual variable for variable lower bounds
......$snx Dual variable of conic constraints
......$bars Dual variable of semidenite domains
......$pobjval Primal objective value
27
......$dobjval Dual objective value
27
......$pobjbound Best primal objective bound from relaxations
27
......$maxinfeas Maximal solution infeasibilities
27
........$pbound In primal inequality constraints
........$peq In primal equality constraints
........$pcone In primal cone constraints
........$dbound In dual inequality constraints
........$deq In dual equality constraints
........$dcone In dual cone constraints
27
Only available if requested by option soldetail.
72
COMMAND REFERENCE
........$int In integer variables
..$iinfo/$dinfo MOSEK information list
28
....$<MSK_INFO> Value of any <MSK_INFO>
The response can be used to determine whether a function call returned successfully.
It contains a numeric response code, code, and a string response message, msg, both
explained in Errors, warnings and response codes in Section 2.2.1. Please also see Section
2.2.5 on how to use this information.
The solution itr will exist whenever the interior-point algorithm has been executed.
This algorithm is used by default for all continuous optimization problems, and has been
described in Section 2.2, 2.3 and 2.4.
The solution bas will exist whenever the interior-point algorithm or the simplex method
has been executed to solve a linear optimization problem (see Section 2.2).
The solution int will exist whenever variables are required to take integer values and
the corresponding Mixed Integer Program has been solved (see Section 2.5).
The iinfo and dinfo lists hold information on the MOSEK optimization library
internals, and have been described in Section 2.8.4.
mosek_clean()
Forces the early release of any previously acquired MOSEK license. If you do not share
a limited number of licenses among multiple users, you do not need to use this function.
The acquisition of a new MOSEK license will automatically take place at the next call to
the function mosek given a valid problem description, using a small amount of extra time.
This usage is briey mentioned in Section 2.1.
For advanced users: If you utilize the .Call convention directly, bypassing the mosek R-
function denition, an Rf_error will result in an unclean memory space. For this reason
you can also use mosek_clean to tidy up uncleaned resources in case an error occurs.
Otherwise this cleaning will not happen until the next call to mosek or until the library is
unloaded. This usage have not been documented elsewhere.
r <- mosek_write(problem, filepath, opts = list())
Outputs a model of an optimization problem in any standard modeling leformat (e.g. lp,
opf, mps, task, etc.), controlled by a set of options. The modeling leformat is selected
based on the extension of the modelle. This function is used and explained in Section 2.7.
28
Only available if requested by option getinfo.
73
COMMAND REFERENCE
The input variable problem could have any name, but should be a list object describing
the optimization problem using the same elds as for the mosek function on page 70.
The input variable lepath should be a string describing the path to modelle. This
path can either be absolute or relative to the working directory, and will overwrite any
existing data on this location. The specied location will be the destination of the exported
model.
The options opts could have any name, and are, in fact, often specied directly as an
anonymous list. It has the following elds.
opts Options for the mosek_write function
..$verbose Output logging verbosity
..$usesol Whether to write an initial solution
..$useparam Whether to write all parameter settings
..$getinfo Whether to extract MOSEK information items
..$scofile Destination of operators from scopt
The argument verbose is used to specify the amount of logging information given by
the interface. This is described in Section 2.6.
The argument usesol can be used to ignore the problem eld sol, while useparam
can be used to ignore the elds iparam, dparam and sparam, when writing the optimization
model. By default, the argument usesol is true and useparam is false, in contrary to the
defaults of the mosek function. These two options are covered in Section 2.7.
The argument getinfo is used to indicate whether to extract information about the
MOSEK optimization library internals. This is described in Section 2.8.4.
The argument scole is used in separable convex optimization to specify the le to
which the operators must be saved. This is described in Section 3.3.3.
The resulting function output variable r, returned by the interface, holds the response of
the function call.
r Result of the mosek_write function
..$response Response from the MOSEK optimization library
....$code ID-code of response
....$msg Human-readable message
..$iinfo/$dinfo MOSEK information list
29
....$<MSK_INFO> Value of any <MSK_INFO>
29
Only available if requested by option getinfo.
74
COMMAND REFERENCE
The response can be used to determined whether a function call returned successfully.
It contains a numeric response code, code, and a string response message, msg, both
explained in Errors, warnings and response codes in Section 2.2.1. Please also see Section
2.7 for an example on how to use this information.
The iinfo and dinfo lists hold information on the MOSEK optimization library
internals, and have been described in Section 2.8.4.
r <- mosek_read(filepath, opts = list())
Interprets a model from any standard modeling leformat (e.g. lp, opf, mps, task, etc.),
controlled by a set of options. The result contains an optimization problem which is
compliant with the input specications of function mosek. This function is used and
explained in Section 2.7.
The input variable lepath should be a string describing the path to le. This path
can either be absolute or relative to the working directory. The specied location will be
the source of the optimization model to be read.
The options opts could have any name, and are, in fact, often input directly as an
anonymous list. It has the following elds.
opts Options for the mosek_read function
..$verbose Output logging verbosity
..$usesol Whether to write an initial solution
..$useparam Whether to write all parameter settings
..$getinfo Whether to extract MOSEK information items
..$scofile Source of operators read to scopt
..$matrixformat The sparse format of the constraint matrix
The argument verbose is used to specify the amount of logging information given by
the interface. This is described in Section 2.6.
The argument usesol can be used to ignore the problem eld sol, while useparam
can be used to ignore the elds iparam, dparam and sparam, when reading the optimization
model. By default, the argument usesol is true and useparam is false, in contrary to the
defaults of the mosek function. These two options are covered in Section 2.7.
The argument getinfo is used to indicate whether to extract information about the
MOSEK optimization library internals. This is described in Section 2.8.4.
75
COMMAND REFERENCE
The argument scole is used in separable convex optimization to specify the le from
which the operators should be read. This is described in Section 3.3.3.
The argument matrixformat determines the matrix format of the problem eld A.
Currently a sparse coordinate (COO) and a compressed sparse column (CSC) format is
supported from package Matrix, along with a simple list-based coordinate format. This
is described in Section 2.7.
The resulting function output variable r, returned by the interface, holds the response of
the function call.
r Result of the mosek_read function
..$response Response from the MOSEK optimization library
....$code ID-code of response
....$msg Human-readable message
..$prob Problem description
..$iinfo/$dinfo MOSEK information list
30
....$<MSK_INFO> Value of any <MSK_INFO>
The response can be used to determined whether a function call returned successfully.
It contains a numeric response code, code, and a string response message, msg, both
explained in Errors, warnings and response codes in Section 2.2.1. Please see Section 2.7
for an example on how to use this information.
The output variable prob is a list object describing the imported optimization problem.
It has the same elds as the input variable problem for the mosek function on page 70.
The iinfo and dinfo lists hold information on the MOSEK optimization library
internals, and have been described in Section 2.8.4.
30
Only available if requested by option getinfo.
76
CONIC TRANSFORMATIONS
B Conic transformations
This appendix will introduce the reader to some of the transformations that make it
possible to formulate a large class of non-linear problems as Conic Quadratic Programs
31
.
Transformations do not have to be hard, and some non-linear constraints easily take the
shape of a quadratic cone (see Section 2.3.1 for denitions). Linear constraints can be
added to a Conic Quadratic Program directly without transformations.
The

x Example Given constraint

x t with x, t 0, we rewrite it to a rotated
quadratic cone as follows:

x t
x t
2
2xr t
2
(B.1)
with linear relationship
r = 1/2 . (B.2)

The denition of linear relationships hardly have any eect on speed of the optimization
process, and many techniques are implemented in MOSEK to eectively reduce the size
of the problem. Thus many CQP formulations solve faster than what could be expected
from the larger model that results from almost any transformation.
The x
3/2
Example Given constraint x
3/2
t with x, t 0, we rewrite it to a pair of
rotated quadratic cones and three linear relationship as follows. Note that from line ve
to six we use the results of the

x example above.
x
3/2
t
x
2 1/2
t
x
2

xt
x
2
2st, 2s

x
x
2
2st, w

v, w = 2s, v = x
x
2
2st, w
2
2vr, w = 2s, v = x, r = 1/2
x
2
2st, w
2
2vr, w = s, v = x, r = 1/8 .
(B.3)

31
More examples and transformation rules can be found online:
Check out: mosek.com Documentation Optimization tools manual Modeling Conic
optimization.
77
CONIC TRANSFORMATIONS
Monomials in General The crucial step in transforming the individual terms (mono-
mials) of a polynomial function to the form of a conic program, is the following recursion
holding true for positive integers n and non-negative y
j
variables. Here, each line implies
the next, and the inequality ends up having the same form as to begin with, only with
n reduced by one. Repeating this procedure until n = 1, the inequality will nally take
the form of a rotated quadratic cone. From the rst expression, we move the coecients
around to get line two, substitute variables by the cones in (B.5) to get line three, and take
the square root on both sides to reach line four.
s
2
n
2
n2
n1
[y
1
y
2
y
2
n]
s
2
n
2
(n1)2
n1
_
(2y
1
y
2
) (2y
3
y
4
) (2y
(2
n
1)
y
2
n)
_
s
2
n
2
(n1)2
n1
_
x
2
1
x
2
2
x
2
2
n1

s
2
(n1)
2
(n1)2
n2
[x
1
x
2
x
2
n1]
(B.4)
Also note that the denition of new variables created by this recursion all takes the form
of a rotated quadratic cone as shown below, with all y
j
0.
x
2
j
2 y
(2j1)
y
2j
j (B.5)
Type I Having the inequality x
p/q
t for some rational exponent p/q 1, we are now
able to transform it into a set of cones by substituting p = 2
n
w for positive integers n
and w. This will yield a form close to the top of (B.4) as shown below, where the variables
x and t just needs to be coupled with s and y for the recursion to be usable. An example
of this follows in (B.9).
x
p/q
t
x
(2
n
w)/q
t
x
2
n
x
w
t
q
(B.6)
Type II The inequality x
p/q
t for some rational exponent 0 p/q 1 and x 0, can
be transformed into the form of a type I monomial quite easily.
x
p/q
t
x t
q/p
(B.7)
Type III Having instead the inequality x
p/q
t for any integers p 1, q 1, we can
use the transform shown below. This again will yield a form close to the top of (B.4),
where s is a constant and the variables just need to be coupled with y. You need to choose
n such that 2
n
p + q in order to make this transformation work. See example (B.10).
x
p/q
t
1 x
p
t
q
(B.8)
78
CONIC TRANSFORMATIONS
Monomials - Example 1 Given x
5/3
t, we rewrite it as shown below.
x
5/3
t
x
5
t
3
x
8
x
3
t
3
s
8
2
12
y
1
y
2
y
8
s = x, y
1
= y
2
= y
3
= x, y
4
= y
5
= y
6
= t, y
7
= 2
12
, y
8
= 1
(B.9)
Monomials - Example 2 Given x
5/2
t, we rewrite it as shown below.
x
5/2
t
x
5
t
2
1 x
5
t
2
s
8
2
12
y
1
y
2
y
8
s = (2
12
)
1/8
, y
1
= y
2
= . . . = y
5
= x, y
6
= y
7
= t, y
8
= 1
(B.10)
Polynomials Now that we have seen how to transform the individual terms (monomials)
of a polynomial function to the form of a conic program, it is easy to transform entire
polynomial constraints. We assume a
j
0 for monomials of type I and III, and a
j
0 for
monomials of type II.

n
j=1
a
j
x
p
j
/q
j
j
b
(B.11)
Substituting each monomial by a new variable u
j
, we are able to isolate each monomial by
itself and use the previously dened transformation rules.

n
j=1
a
j
u
j
b
x
p
j
/q
j
j
u
j
j of type I and III
x
p
j
/q
j
j
u
j
j of type II
(B.12)
Regarding the objective function, minimization problems can be rewritten as shown below
matching (B.11) and, therefore, it is applicable to the isolation of monomials and the whole
transformation used above. In this way the objective function can also take the linear form
required by conic programs.
minimize f(x)
minimize b
f(x) b
(B.13)
79
CONIC TRANSFORMATIONS
Polynomials - Example 1 This is a problem with type III monomials
minimize c
T
x
subject to

n
j=1
f
j
x
j
b
x 0
(B.14)
where it is assumed that f
j
> 0 and b > 0. It is equivalent to
minimize c
T
x
subject to

n
j=1
f
j
z
j
= b
v
j
=

2 j = 1, . . . , n
v
2
j
2z
j
x
j
j = 1, . . . , n
x, z 0
(B.15)
Polynomials - Example 2 The objective function with a mixture of type I and type
III monomials
minimize x
2
+ x
2
(B.16)
is used in statistical matching and can be formulated as
minimize u + v
subject to x
2
u
x
2
v
(B.17)
which is equivalent to the quadratic conic optimization problem
minimize u + v
subject to x
2
2uw
s
2
2y
21
y
22
y
2
21
2y
1
y
2
y
2
22
2y
3
y
4
w = 1
s = 2
3/4
y
1
= y
2
= x
y
3
= v
y
4
= 1
(B.18)
80
B.1 The Quadratic Program CONIC TRANSFORMATIONS
B.1 The Quadratic Program
Any convex quadratic problem can be stated on the form
minimize 0.5|Fx|
2
+ c
T
x
subject to 0.5|Gx|
2
+ a
T
x b
(B.19)
where F and G are matrices and c and a are column vectors. Here a convex quadratic
term x
T
Qx would be written |Fx|
2
, where F
T
F = Q is the Cholesky factorization. For
simplicitys sake we assume that there is only one constraint, but it should be obvious how
to generalize the methods to an arbitrary number of constraints. Problem (B.19) can be
reformulated as
minimize 0.5|t|
2
+ c
T
x
subject to 0.5|z|
2
+ a
T
x b
Fx t = 0
Gx z = 0
(B.20)
after the introduction of the new variables t and z. It is easy to convert this problem to a
conic quadratic optimization problem, i.e.
minimize v + c
T
x
subject to p + a
T
x = b
Fx t = 0
Gx z = 0
w = 1
q = 1
|t|
2
2vw v, w 0
|z|
2
2pq p, q 0
(B.21)
In this case the last two inequalities both take the form of a rotated quadratic cone, and
the entire problem can be solved as a conic quadratic program, Section 2.3, using the
interior-point algorithm.
If F is a non-singular matrix - e.g. a diagonal matrix - then x can be eliminated from
the problem using the substitution x = F
1
t. In most cases the MOSEK optimization
library will perform this reduction automatically during the presolve phase before the
actual optimization is performed.
81