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

Akatsuki Nishioka, Mitsuru Toyoda, Mirai Tanaka, Yoshihiro Kanno

TL;DR

This paper analyzes the problem of minimizing the largest generalized eigenvalue lambda_1^{A,B}(x) for symmetric-matrix-valued affine functions with B(x) ≻ 0 over a compact convex set. It derives an explicit Clarke subdifferential, proves pseudoconvexity under reasonable assumptions, and develops a smoothing-based projected gradient framework with a proven O(k^{-1/2}) convergence rate to a global optimum, complemented by practical acceleration and inexact smoothing strategies. The approach is demonstrated on eigenfrequency optimization of truss structures, with x_min > 0 ensuring non-singularity, and shows that acceleration and inexact smoothing can substantially reduce computational costs while preserving solution quality. The results provide a theoretically grounded avenue for large-scale generalized eigenvalue optimization and offer intuition for extending to more complex matrix-valued nonlinearities and constraints.

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.

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

TL;DR

This paper analyzes the problem of minimizing the largest generalized eigenvalue lambda_1^{A,B}(x) for symmetric-matrix-valued affine functions with B(x) ≻ 0 over a compact convex set. It derives an explicit Clarke subdifferential, proves pseudoconvexity under reasonable assumptions, and develops a smoothing-based projected gradient framework with a proven O(k^{-1/2}) convergence rate to a global optimum, complemented by practical acceleration and inexact smoothing strategies. The approach is demonstrated on eigenfrequency optimization of truss structures, with x_min > 0 ensuring non-singularity, and shows that acceleration and inexact smoothing can substantially reduce computational costs while preserving solution quality. The results provide a theoretically grounded avenue for large-scale generalized eigenvalue optimization and offer intuition for extending to more complex matrix-valued nonlinearities and constraints.

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.
Paper Structure (30 sections, 12 theorems, 61 equations, 8 figures, 1 table)

This paper contains 30 sections, 12 theorems, 61 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivating example
  3. Related work
  4. Contribution
  5. Paper organization and notation
  6. Preliminaries
  7. Definitions
  8. Problem setting
  9. Application: eigenfrequency optimization of truss structures
  10. Basic properties and problem classes
  11. Properties of the maximum generalized eigenvalue
  12. Transformation into standard eigenvalues
  13. The Clarke subdifferential
  14. Pseudoconvexity
  15. Smoothing method
  16. ...and 15 more sections

Key Result

Lemma 2

Let $A,B:\mathbb{R}^m_{>0}\to\mathbb{S}^n$ be continuously differentiable symmetric-matrix-valued functions such that $B(\bm{x})\succ 0$ for any $\bm{x}\in\mathbb{R}^m_{>0}$, and $B(\bm{x})^{1/2}$ be the positive definite square root of $B(\bm{x})$. The symmetric-matrix-valued function $B(\bm{x})^{-

Figures (8)

  • Figure 1: Eigenfrequency optimization of a truss structure
  • Figure 4: Difference of the objective value and the optimal value at each iteration (comparison with acceleration)
  • Figure 5: Designs after $3000$ iterations
  • Figure 9: Difference of the objective value and the optimal value at each iteration (comparison with inexact smoothing)
  • Figure 10: Designs after $3000$ iterations
  • ...and 3 more figures

Theorems & Definitions (21)

  • Definition 1: Generalized eigenvalue
  • Definition 2: Clarke subdifferential
  • Definition 3: Clarke stationary point
  • Definition 4: Quasiconvex function
  • Definition 5: Pseudoconvex function
  • Example 1
  • Lemma 2
  • Lemma 3: overton92, Theorem 3
  • Theorem 4: The Clarke subdifferential of the maximum generalized eigenvalue
  • Corollary 5
  • ...and 11 more