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Computational Optimization and Applications

, Volume 58, Issue 2, pp 423–454 | Cite as

The space decomposition theory for a class of eigenvalue optimizations

  • Ming Huang
  • Li-Ping PangEmail author
  • Zun-Quan Xia
Article

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.

Keywords

Nonsmooth optimization Eigenvalue optimization \(\mathcal{VU}\)-Decomposition \({\mathcal{U}}\)-Lagrangian D.C. function Second-order derivative 

Notes

Acknowledgements

We would like to thank Xi-jun Liang and Yue Lu from Dalian University of Technology and Yuan Lu from Shenyang University for numerous fruitful discussions. The authors also thank two anonymous referees for a number of valuable and constructive suggestions that helped to improved the presentation. This research was supported in part by the Natural Science Foundation of China, Grant 11171049, 11226230 and 11301347, and General Project of the Education Department of Liaoning Province L2012427.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.CORA, School of Mathematical SciencesDalian University of TechnologyDalianChina

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