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

Akatsuki Nishioka (), Mitsuru Toyoda, Mirai Tanaka and Yoshihiro Kanno
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Akatsuki Nishioka: The University of Tokyo
Mitsuru Toyoda: Tokyo Metropolitan University
Mirai Tanaka: The Institute of Statistical Mathematics
Yoshihiro Kanno: The University of Tokyo

Computational Optimization and Applications, 2025, vol. 90, issue 1, No 10, 303-336

Abstract: 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.

Keywords: Generalized eigenvalue optimization; Quasiconvex optimization; Pseudoconvex optimization; Structural optimization; Smoothing method; 90C26; 90C90 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-024-00621-4

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