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A semismooth Newton based dual proximal point algorithm for maximum eigenvalue problem

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  • Yong-Jin Liu

    (Fuzhou University)

  • Jing Yu

    (Fuzhou University)

Abstract

The maximum eigenvalue problem is to minimize the maximum eigenvalue function over an affine subspace in a symmetric matrix space, which has many applications in structural engineering, such as combinatorial optimization, control theory and structural design. Based on classical analysis of proximal point (Ppa) algorithm and semismooth analysis of nonseparable spectral operator, we propose an efficient semismooth Newton based dual proximal point (Ssndppa) algorithm to solve the maximum eigenvalue problem, in which an inexact semismooth Newton (Ssn) algorithm is applied to solve inner subproblem of the dual proximal point (d-Ppa) algorithm. Global convergence and locally asymptotically superlinear convergence of the d-Ppa algorithm are established under very mild conditions, and fast superlinear or even quadratic convergence of the Ssn algorithm is obtained when the primal constraint nondegeneracy condition holds for the inner subproblem. Computational costs of the Ssn algorithm for solving the inner subproblem can be reduced by fully exploiting low-rank or high-rank property of a matrix. Numerical experiments on max-cut problems and randomly generated maximum eigenvalue optimization problems demonstrate that the Ssndppa algorithm substantially outperforms the Sdpnal+ solver and several state-of-the-art first-order algorithms.

Suggested Citation

  • Yong-Jin Liu & Jing Yu, 2023. "A semismooth Newton based dual proximal point algorithm for maximum eigenvalue problem," Computational Optimization and Applications, Springer, vol. 85(2), pages 547-582, June.
    • Handle: RePEc:spr:coopap:v:85:y:2023:i:2:d:10.1007_s10589-023-00467-2
    DOI: 10.1007/s10589-023-00467-2
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    References listed on IDEAS

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    1. Radu Ioan Bot & Dang-Khoa Nguyen, 2020. "The Proximal Alternating Direction Method of Multipliers in the Nonconvex Setting: Convergence Analysis and Rates," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 682-712, May.
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    4. Yong-Jin Liu & Jing Yu, 2022. "A Semismooth Newton-based Augmented Lagrangian Algorithm for Density Matrix Least Squares Problems," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 749-779, December.
    5. Xin Chen & Houduo Qi & Liqun Qi & Kok-Lay Teo, 2004. "Smooth Convex Approximation to the Maximum Eigenvalue Function," Journal of Global Optimization, Springer, vol. 30(2), pages 253-270, November.
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