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A space decomposition scheme for maximum eigenvalue functions and its applications

Author

Listed:
  • Ming Huang

    (Dalian University of Technology
    Dalian Maritime University)

  • Yue Lu

    (Tianjin Normal University)

  • Li Ping Pang

    (Dalian University of Technology)

  • Zun Quan Xia

    (Dalian University of Technology)

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 $${\mathcal {U}}$$ 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 $${\mathcal {U}}$$ 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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:85:y:2017:i:3:d:10.1007_s00186-017-0579-z
    DOI: 10.1007/s00186-017-0579-z
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    References listed on IDEAS

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    1. A. S. Lewis, 1996. "Derivatives of Spectral Functions," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 576-588, August.
    2. 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.
    3. 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.
    4. Gábor Pataki, 1998. "On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 339-358, May.
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