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A space decomposition scheme for maximum eigenvalue functions and its applications

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Abstract

In this paper, we study nonlinear optimization problems involving eigenvalues of symmetric matrices. One of the difficulties in solving these problems is that the eigenvalue functions are not differentiable when the multiplicity of the function is not one. We apply the \({\mathcal {U}}\)-Lagrangian theory to analyze the largest eigenvalue function of a convex matrix-valued mapping which extends the corresponding results for linear mapping in the literature. We also provides the formula of first-and second-order derivatives of the \({\mathcal {U}}\)-Lagrangian under mild assumptions. These theoretical results provide us new second-order information about the largest eigenvalue function along a suitable smooth manifold, and leads to a new algorithmic framework for analyzing the underlying optimization problem.

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Acknowledgements

The authors would thank the associate editor and the anonymous referees for their valuable and helpful comments and good advice for improving the presentation of this paper. This article is supported by the National Natural Science Foundation of China under Grant Nos. 11301347, 11626053 and 11601389, the Project funded by China Postdoctoral Science Foundation Under No. 2016M601296 and the Fundamental Research Funds for the Central Universities under Project No. 3132016108, the Scientific Research Foundation Funds of DLMU under Project No. 02501102, and Doctoral Foundation of Tianjin Normal University under Project No. 52XB1513.

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Authors and Affiliations

  1. School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024, China

    Ming Huang

  2. Department of Mathematics, Dalian Maritime University, Dalian, 116026, China

    Ming Huang

  3. School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China

    Yue Lu

  4. CORA, School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Li Ping Pang & Zun Quan Xia

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  1. Ming Huang
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  2. Yue Lu
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  3. Li Ping Pang
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  4. Zun Quan Xia
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Correspondence to Ming Huang.

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A correction to this article is available online at https://doi.org/10.1007/s00186-017-0622-0.

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Huang, M., Lu, Y., Pang, L.P. et al. A space decomposition scheme for maximum eigenvalue functions and its applications. Math Meth Oper Res 85, 453–490 (2017). https://doi.org/10.1007/s00186-017-0579-z

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  • DOI: https://doi.org/10.1007/s00186-017-0579-z

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