Abstract
Numerically stable algorithms for quadratic programming are discussed. A new algorithm is described for indefinite quadratic programming which utilizes methods for updating positivedefinite factorizations only. Consequently all the updating procedures required are common to algorithms for linearly-constrained optimization. The new algorithm can be used for the positive-definite case without loss of efficiency.
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References
P. Businger and G.H. Golub, “Linear least squares solutions by Householder transformations”,Numerische Mathematik 7 (1965) 269–276.
A.R. Conn, “Constrained optimization using a non-differentiable penalty function”,SIAM Journal of the Numerical Analysis 10 (1973) 760–784.
A.R. Conn and J.W. Sinclair, “Quadratic programming via a non-differentiable penalty function”, Department of Combinations and Optimization Research, University of Waterloo, Rep. CORR 75-15 (1975).
R. Fletcher, “The calculation of feasible points for linearly constrained optimization problems”, UKAEA Research Group Rept., AERE R6354 (1970).
R. Fletcher and M.P. Jackson, “Minimization of a quadratic function of many variables subject only to lower and upper bounds”,Journal of the Institute of Mathematics and its Applications 14 (1974) 159–174.
P.E. Gill, “Numerical methods for large-scale linearly-constrained optimization problems”, Ph.D. thesis, Imperial College of Science and Technology, London University (1975).
P.E. Gill and W. Murray, “Quasi-Newton methods for linearly-constrained optimization”, in: P.E. Gill and W. Murray, eds.,Numerical methods for constrained optimization (Academic Press, London, 1974) pp. 67–92.
P.E. Gill and W. Murray, “Newton-type methods for linearly-constrained optimization”, in: P.E. Gill and W. Murray, eds.,Numerical methods for constrained optimization (Academic Press, London, 1974) pp. 29–66.
P.E. Gill and W. Murray, “Newton-type methods for unconstrained and linearly-constrained optimization”,Mathematical Programming 7 (1974) 311–350.
P.E. Gill and W. Murray, “Minimization of a nonlinear function subject to bounds on the variables”, NPL NAC Rep. No. 72 (1976).
P.E. Gill and W. Murray, “The computation of Lagrange-multiplier estimates for constrained minimization”, NPL NAC Rep. No. 77 (1977).
P.E. Gill and W. Murray, “Linearly-constrained problems including linear and quadratic programming”, in: D. Jacobs, ed.,A survey of numerical analysis—1976, (Academic Press, London, 1977).
P.E. Gill, G.H. Golub, W. Murray and M.A. Saunders, “Methods for modifying matrix factorizations”,Mathematics of Computation 28 (1974) 505–535.
C.L. Lawson and R.J. Hanson,Solving least-squares problems (Prentice Hall, Englewood Cliffs, NJ, 1974).
W. Murray, “An algorithm for finding a local minimum of an indefinite quadratic program”, NPL NAC Rep. No. 1 (1971).
J.H. Wilkinson,The algebraic eigenvalue problem (Oxford University Press, London, 1965).
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Gill, P.E., Murray, W. Numerically stable methods for quadratic programming. Mathematical Programming 14, 349–372 (1978). https://doi.org/10.1007/BF01588976
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DOI: https://doi.org/10.1007/BF01588976