A PRECONDITIONING FRAMEWORK FOR SEQUENCES OF DIAGONALLY MODIFIED LINEAR SYSTEMS ARISING IN OPTIMIZATION

We propose a framework for building preconditioned for sequences of linear systems of the form (A + ∆ k )x k = b k , where A is symmetric positive semidefinite and ∆ k is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-c...

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Bibliographic Details
Published in:SIAM journal on numerical analysis Vol. 50; no. 6; pp. 3280 - 3302
Main Authors: BELLAVIA, STEFANIA, DE SIMONE, VALENTINA, DI SERAFINO, DANIELA, MORINI, BENEDETTA
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
Subjects:
Algorithms
Convex analysis
Eigenvalues
Factorization
Hessian matrices
Linear systems
Mathematical sequences
Matrices
Methods
Newtons method
Numerical analysis
Optimization
Preconditioning
Quadratic programming
ISSN:0036-1429, 1095-7170
Online Access:Get full text
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Summary:We propose a framework for building preconditioned for sequences of linear systems of the form (A + ∆ k )x k = b k , where A is symmetric positive semidefinite and ∆ k is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region and overestimation methods for convex unconstrained optimization problems and nonlinear least squares. For all the matrices of a sequence, the preconditioners are obtained by updating any preconditioner for A available in the LDL T form. The preconditioners in the framework satisfy the natural requirement of being effective on slowly varying sequences; furthermore, under an additional property they are also able to cluster eigenvalues of the preconditioned matrix when some entries of ∆ k are sufficiently large. We present two low-cost preconditioners sharing the above-mentioned properties and evaluate them on sequences of linear systems generated by the reflective Newton method applied to bound-constrained convex quadratic programming problems and on sequences arising in solving nonlinear least-squares problems with the regularized Euclidean residual method. The results of the numerical experiments show the effectiveness of these preconditioners.
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ISSN:0036-1429
1095-7170
DOI:10.1137/110860707