The space decomposition theory for a class of eigenvalue optimizations

Ming Huang (), Li-Ping Pang () and Zun-Quan Xia ()

Computational Optimization and Applications, 2014, vol. 58, issue 2, 423-454

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

Keywords: Nonsmooth optimization; Eigenvalue optimization; $\mathcal{VU}$ -Decomposition; ${\mathcal{U}}$ -Lagrangian; D.C. function; Second-order derivative (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10589-013-9624-x

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