A D.C. Optimization Algorithm for Solving the Trust-Region Subproblem

This paper is devoted to difference of convex functions (d.c.) optimization: d.c. duality, local and global optimality conditions in d.c. programming, the d.c. algorithm (DCA), and its application to solving the trust-region problem. The DCA is an iterative method that is quite different from well-k...

Full description

Saved in:
Bibliographic Details
Published in:SIAM journal on optimization Vol. 8; no. 2; pp. 476 - 505
Main Authors: Tao, Pham Dinh, An, Le Thi Hoai
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.05.1998
Subjects:
Computer Science
Convex analysis
Eigenvalues
Eigenvectors
Methods
Numerical analysis
Optimization algorithms
Simulation
ISSN:1052-6234, 1095-7189
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:This paper is devoted to difference of convex functions (d.c.) optimization: d.c. duality, local and global optimality conditions in d.c. programming, the d.c. algorithm (DCA), and its application to solving the trust-region problem. The DCA is an iterative method that is quite different from well-known related algorithms. Thanks to the particular structure of the trust-region problem, the DCA is very simple (requiring only matrix-vector products) and, in practice, converges to the global solution. The inexpensive implicitly restarted Lanczos method of Sorensen is used to check the optimality of solutions provided by the DCA. When a nonglobal solution is found, a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the DCA at this point. It is shown that in the nonconvex case, the DCA converges to the global solution of the trust-region problem, using only matrix-vector products and requiring at most 2m+2 restarts, where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem. Numerical simulations establish the robustness and efficiency of the DCA compared to standard related methods, especially for large-scale problems.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:1052-6234
1095-7189
DOI:10.1137/S1052623494274313