Sum-of-squares relaxations for polynomial min–max problems over simple sets

We consider min–max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables. When the variables belong to simple sets (e.g., a hypercube, t...

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Bibliographic Details
Published in:Mathematical programming Vol. 209; no. 1; pp. 475 - 501
Main Author: Bach, Francis
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2025
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
11E25
Semidefinite programming
Sum-of-squares
Polynomial optimization
90C22
Min–max problems
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We consider min–max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables. When the variables belong to simple sets (e.g., a hypercube, the Euclidean hypersphere, or a ball), we derive a sum-of-squares formulation based on a primal-dual approach. In the simplest setting, we provide a convergence proof when the degree of the relaxation tends to infinity and observe empirically that it can be finitely convergent in several situations. Moreover, our formulation leads to an interesting link with feasibility certificates for polynomial inequalities based on Putinar’s Positivstellensatz.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-024-02072-5