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Eigenvalue programming beyond matrices

Author

Listed:
  • Masaru Ito

    (Nihon University)

  • Bruno F. Lourenço

    (Institute of Statistical Mathematics)

Abstract

In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the “eigenvalues” of the decision variable. Here, by making use of the FTvN systems framework introduced by Gowda, we interpret “eigenvalues” in a broad fashion going beyond the usual eigenvalues of matrices. This allows us to shed new light on classical problems such as inverse eigenvalue problems and also leads to new applications. In particular, after analyzing and developing a simple projected gradient algorithm for general eigenvalue programs, we show that eigenvalue programs can be used to express what we call vanishing quadratic constraints. A vanishing quadratic constraint requires that a given system of convex quadratic inequalities be satisfied and at least a certain number of those inequalities must be tight. As a particular case, this includes the problem of finding a point x in the intersection of m ellipsoids in such a way that x is also in the boundary of at least $$\ell $$ ℓ of the ellipsoids, for some fixed $$\ell > 0$$ ℓ > 0 . At the end, we also present some numerical experiments.

Suggested Citation

  • Masaru Ito & Bruno F. Lourenço, 2024. "Eigenvalue programming beyond matrices," Computational Optimization and Applications, Springer, vol. 89(2), pages 361-384, November.
    • Handle: RePEc:spr:coopap:v:89:y:2024:i:2:d:10.1007_s10589-024-00591-7
    DOI: 10.1007/s10589-024-00591-7
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    References listed on IDEAS

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    1. Pierre Frankel & Guillaume Garrigos & Juan Peypouquet, 2015. "Splitting Methods with Variable Metric for Kurdyka–Łojasiewicz Functions and General Convergence Rates," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 874-900, June.
    2. Adrian S. Lewis & Jérôme Malick, 2008. "Alternating Projections on Manifolds," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 216-234, February.
    3. Chen Chen & Ting Kei Pong & Lulin Tan & Liaoyuan Zeng, 2020. "A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection," Journal of Global Optimization, Springer, vol. 78(1), pages 107-136, September.
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