Computing local minimizers in polynomial optimization under genericity conditions Computing local minimizers in polynomial optimization under genericity conditions

In this paper, we focus on computing local minimizers of a multivariate polynomial optimization problem under certain genericity conditions. Using a technique from computer algebra and the second-order optimality condition, we provide a univariate representation for the set of local minimizers. In p...

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Bibliographic Details
Published in:Journal of global optimization Vol. 92; no. 4; pp. 909 - 932
Main Authors: Hieu, Vu Trung, Takeda, Akiko
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2025
Springer Nature B.V
Subjects:
Algebra
Algorithms
Computation
Computer algebra
Computer Science
Constraints
Deep learning
Linear programming
Machine learning
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Perturbation methods
Polynomial matrices
Polynomials
Real Functions
Local minimizer
Polynomial optimization
Reduced Gröbner basis
Radical of an ideal
13P10
90C23
Second-order optimality condition
14P05
Zero-dimensional ideal
Symbolic algorithm
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary:In this paper, we focus on computing local minimizers of a multivariate polynomial optimization problem under certain genericity conditions. Using a technique from computer algebra and the second-order optimality condition, we provide a univariate representation for the set of local minimizers. In particular, for the unconstrained problem, i.e., the constraint set is R n , the coordinates of all local minimizers can be represented by the values of n univariate polynomials at the real solutions of a univariate system containing a polynomial equation and a polynomial matrix inequality. We also develop the technique for problems with equality/inequality constraints. Based on the above technique, we design algorithms to enumerate the local minimizers and provide some experimental examples based on hybrid symbolic-numerical computations. For the case that the genericity conditions fail, at the end of the paper we propose a perturbation technique to compute approximately a global minimizer, provided that the constraint set is compact.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-025-01500-w