MatLab Optimization tool box
S. M. Shafkat Raihan
December 6, 2020
Exercise 1
Question:
By taking account of any optimization problem demonstrate the use of various functions
available in the MatLab optimization Toolbox.
Answer:
The Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock
in 1960, which is used as a performance test problem for optimization algorithms. It is also
known as Rosenbrock’s valley or Rosenbrock’s banana function. The global minimum is inside
a long, narrow, parabolic shaped flat valley. To find the valley is trivial. To converge to the
global minimum, however, is difficult. The function is defined by
𝑓 (𝑥, 𝑦) = (𝑎 − 𝑥)2 + 𝑏(𝑦 − 𝑥 2 )2
It has a global minimum at (𝑥, 𝑦) = (𝑎, 𝑎2 ), where f (𝑥, 𝑦) = 0. Usually these parameters are set
such that 𝑎 = 1, 𝑏 = 100. Only in the trivial case where 𝑎 = 0 the function is symmetric and the
minimum is at the origin. To optimize it, we apply various MATLAB optimization functions.
Unconstrained minimization: Unconstrained minimization - fminunc,for unconstrained
multivariable function
%Optimizing Rosenbrock function
%Unconstrained minimization - fminunc(for unconstrained multivariable
%function),fminsearch(for unconstrained multivariable function using derivative-f
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
x0 = [1,1];
[x,fval] = fminunc(fun,x0);
fprintf(’Unconstrained Minimum is at %f,%f \n’,x(1),x(2))
Unconstrained Minimum is at 1.000000,1.000000.
Unconstrained minimization: Unconstrained minimization - fminsearch, for uncon-
strained multivariable function using derivative-free method.
fun = @(x)100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;
x0 = [-1.2,1];
1
x = fminsearch(fun,x0);
fprintf(’Unconstrained Minimum with derivative-free method is %f,%f \n’,x(1),x(2)
Unconstrained Minimum with derivative-free method is 1.000022,1.000042
Constrained Minimization
%constrained nonlinear multivariable problems
% With linear constraints
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2; %Anonymous function used to define ob
x0 = [0.5,0]; %Initial point
A = [1,2]; % A is a matrix which needed for the inequality constraint A.x <= b
b = 1; % b is vector which needed for the constraint A.x <= b
Aeq = [2,1]; % Aeq is a matrix which needed for the equality constraint Aeq.x ==
beq = 1; % beq is a vector which needed for the equality constraint Aeq.x == beq
x = fmincon(fun,x0,A,b,Aeq,beq);
fprintf(’Minimum with linear constraints at %f,%f\n’,x(1),x(2))
Minimum with linear constraints at 0.414944,0.170111
%With bound constraints
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
%bound constraints are 0 <= x(1) <= 1, 0 <= x(2) <= 2
lb = [0,0]; %lower bound
ub = [1,2]; %upperbound
A = [];
b = [];
Aeq = [];
beq = [];
x0 = (lb + ub)/2; % start of in the middle point of [lb,ub]
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub);
fprintf(’Minimum with bound constraints at %f,%f\n’,x(1),x(2))
Minimum with bound constraints at 0.999640,0.999280
%With nonlinear constraints
%Find the point where Rosenbrock’s function is minimized
%within a circle, also subject to bound constraints
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
%bound constraints are 0 <= x(1) <= 0.5, 0.2 <= x(2) <= 0.8
lb = [0,0.2]; %lower bound
ub = [0.5,0.8]; %upper bound
% Also look within an ellipse centered at [1/3,1/3] with
% major radius a = 1. and minor radius b = 2/3
nonlcon = @ellipsecon;
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon);
fprintf(’Minimum with nonlinear constraints at %f,%f\n’,x(1),x(2))
%Minimum with nonlinear constraints at 0.500000,0.250000
function [c,ceq] = ellipsecon(x)
c = (x(1)-1/3)^2 + 9*((x(2)-1/3)^2)/4 - 1;
ceq = [];
2
end
Minimum with nonlinear constraints at 0.500000,0.250000
