Computing local minimizers in polynomial optimization under genericity conditions

Vu Trung Hieu () and Akiko Takeda ()
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Vu Trung Hieu: Norwegian University of Science and Technology
Akiko Takeda: Center for Advanced Intelligence Project, RIKEN

Journal of Global Optimization, 2025, vol. 92, issue 4, No 3, 909-932

Abstract: Abstract 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 $${{\,\mathrm{\mathbb {R}}\,}}^n$$ 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.

Keywords: Polynomial optimization; Local minimizer; Second-order optimality condition; Zero-dimensional ideal; Radical of an ideal; Reduced Gröbner basis; Symbolic algorithm; 14P05; 13P10; 90C23 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10898-025-01500-w

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