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On a minimization problem of the maximum generalized eigenvalue: properties and algorithms

Author

Listed:
  • Akatsuki Nishioka

    (The University of Tokyo)

  • Mitsuru Toyoda

    (Tokyo Metropolitan University)

  • Mirai Tanaka

    (The Institute of Statistical Mathematics
    RIKEN Center for Advanced Intelligence Project)

  • Yoshihiro Kanno

    (The University of Tokyo
    The University of Tokyo)

Abstract

We study properties and algorithms of a minimization problem of the maximum generalized eigenvalue of symmetric-matrix-valued affine functions, which is nonsmooth and quasiconvex, and has application to eigenfrequency optimization of truss structures. We derive an explicit formula of the Clarke subdifferential of the maximum generalized eigenvalue and prove the maximum generalized eigenvalue is a pseudoconvex function, which is a subclass of a quasiconvex function, under suitable assumptions. Then, we consider smoothing methods to solve the problem. We introduce a smooth approximation of the maximum generalized eigenvalue and prove the convergence rate of the smoothing projected gradient method to a global optimal solution in the considered problem. Also, some heuristic techniques to reduce the computational costs, acceleration and inexact smoothing, are proposed and evaluated by numerical experiments.

Suggested Citation

  • Akatsuki Nishioka & Mitsuru Toyoda & Mirai Tanaka & Yoshihiro Kanno, 2025. "On a minimization problem of the maximum generalized eigenvalue: properties and algorithms," Computational Optimization and Applications, Springer, vol. 90(1), pages 303-336, January.
    • Handle: RePEc:spr:coopap:v:90:y:2025:i:1:d:10.1007_s10589-024-00621-4
    DOI: 10.1007/s10589-024-00621-4
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2007. "Smoothing technique and its applications in semidefinite optimization," LIDAM Reprints CORE 1951, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, July.
    3. Quoc Tran-Dinh, 2017. "Adaptive smoothing algorithms for nonsmooth composite convex minimization," Computational Optimization and Applications, Springer, vol. 66(3), pages 425-451, April.
    4. Lv, Jian & Pang, Li-Ping & Wang, Jin-He, 2015. "Special backtracking proximal bundle method for nonconvex maximum eigenvalue optimization," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 635-651.
    5. Nesterov, Y. & Nemirovskii, A., 1995. "An interior-point method for generalized linear-fractional programming," LIDAM Reprints CORE 1168, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Xiaoqi Yang & Chenchen Zu, 2022. "Convergence of Inexact Quasisubgradient Methods with Extrapolation," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 676-703, June.
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