A Fast Space-decomposition Scheme for Nonconvex Eigenvalue Optimization

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• 作者：Ming Huang ; Li-Ping Pang ; Yuan Lu ; Zun-Quan Xia
• 关键词：Second ; order bundle methods ; 𝓥𝓤 ; decomposition ; Nonconvex eigenvalue optimization ; Composite optimization ; Bilinear matrix inequality
• 刊名：Set-Valued and Variational Analysis
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：25
• 期：1
• 页码：43-67
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Probability Theory and Stochastic Processes;
• 出版者：Springer Netherlands
• ISSN：1877-0541
• 卷排序：25

文摘

In this paper, a nonsmooth bundle algorithm to minimize the maximum eigenvalue function of a nonconvex smooth function is presented. The bundle method uses an oracle to compute separately the function and subgradient information for a convex function, and the function and derivative values for the smooth mapping. Using this information, in each iteration, we replace the smooth inner mapping by its Taylor-series linearization around the current serious step. To solve the convex approximate eigenvalue problem with affine mapping faster, we adopt the second-order bundle method based on 𝓥𝓤-decomposition theory. Through the backtracking test, we can make a better approximation for the objective function. Quadratic convergence of our special bundle method is given, under some additional assumptions. Then we apply our method to some particular instance of nonconvex eigenvalue optimization, specifically: bilinear matrix inequality problems.