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The space decomposition theory for a class of eigenvalue optimizations

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  • Ming Huang
  • Li-Ping Pang
  • Zun-Quan Xia

Abstract

In this paper we study optimization problems involving eigenvalues of symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, considered as functions of a symmetric matrix, are not differentiable at those points where they coalesce. Here we apply the $\mathcal{U}$ -Lagrangian theory to a class of D.C. functions (the difference of two convex functions): the arbitrary eigenvalue function λ i , with affine matrix-valued mappings, where λ i is a D.C. function. We give the first-and second-order derivatives of ${\mathcal{U}}$ -Lagrangian in the space of decision variables R m when transversality condition holds. Moreover, an algorithm framework with quadratic convergence is presented. Finally, we present an application: low rank matrix optimization; meanwhile, list its $\mathcal{VU}$ decomposition results. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Ming Huang & Li-Ping Pang & Zun-Quan Xia, 2014. "The space decomposition theory for a class of eigenvalue optimizations," Computational Optimization and Applications, Springer, vol. 58(2), pages 423-454, June.
    • Handle: RePEc:spr:coopap:v:58:y:2014:i:2:p:423-454
    DOI: 10.1007/s10589-013-9624-x
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    References listed on IDEAS

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    1. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    2. C. Helmberg & F. Rendl & R. Weismantel, 2000. "A Semidefinite Programming Approach to the Quadratic Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 4(2), pages 197-215, June.
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    Cited by:

    1. Ming Huang & Yue Lu & Li Ping Pang & Zun Quan Xia, 2017. "A space decomposition scheme for maximum eigenvalue functions and its applications," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(3), pages 453-490, June.

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