A Second-Order Bundle Method Based on -Decomposition Strategy for a Special Class of Eigenvalue Optimizations. (2nd February 2016)

• Record Type: Journal Article
• Authors: Huang, Ming; Pang, Li-Ping; Liang, Xi-Jun; Meng, Fan-Yun
• Is Part Of: Numerical functional analysis and optimization. Volume 37:Number 5(2016)
• Page Start: 554
• Page End: 582
• Publication Date: 2016-02-02
• Subjects: D.C. function -- eigenvalue optimization -- nonconvex optimization -- nonsmooth optimization -- second-order proximal bundle method -- -Lagrangian -- -decomposition; 15A18 -- 65K10 -- 90C26 -- 49J52; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2016.1138969
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: In the past decade, eigenvalue optimization has gained remarkable attention in various engineering applications. One of the main difficulties with numerical analysis of such problems is that the eigenvalues, considered as functions of a symmetric matrix, are not smooth at those points where they are multiple. We propose a new explicit nonsmooth second-order bundle algorithm based on the idea of the proximal bundle method on minimizing the arbitrary eigenvalue over an affine family of symmetric matrices, which is a special class of eigenvalue function–D.C. function. To the best of our knowledge, few methods currently exist for minimizing arbitrary eigenvalue function. In this work, we apply the -Lagrangian theory to this class of D.C. functions: the arbitrary eigenvalue function λ i with affine matrix-valued mappings, where λ i is usually not convex. We prove the global convergence of our method in the sense that every accumulation point of the sequence of iterates is stationary. Moreover, under mild conditions we show that, if started close enough to the minimizer x *, the proposed algorithm converges to x * quadratically. The method is tested on some constrained optimization problems, and some encouraging preliminary numerical results show the efficiency of our method.