A Second-Order Bundle Method Based on UV-Decomposition Strategy for a Special Class of Eigenvalue Optimizations

摘要

In the past decade, eigenvalue optimization has gained remark able 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 pro pose 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 U-Lagrangian theory to this class of D.C. functions: the arbitrary eigenvalue function lambda(i) with affine matrix-valued mappings, where lambda(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.