All saddle points for polynomial optimization

Anwa Zhou (), Shiqian Yin () and Jinyan Fan ()
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Anwa Zhou: Shanghai University
Shiqian Yin: Shanghai University
Jinyan Fan: Shanghai Jiao Tong University

Computational Optimization and Applications, 2025, vol. 90, issue 3, No 5, 752 pages

Abstract: Abstract In this paper, we study how to compute all saddle points for the constrained and unconstrained polynomial optimization, respectively. For the constrained polynomial optimization, a scalar-type semidefinite relaxation algorithm is proposed based on the Karush-Kuhn-Tucker conditions. While for the unconstrained polynomial optimization, a matrix-type semidefinite relaxation algorithm is proposed based on the second-order optimality conditions. Both algorithms can detect the nonexistence of saddle points or find all of them if there are finitely many ones. The finite convergence of the algorithms can also be obtained under some genericity conditions.

Keywords: Saddle points; Polynomial optimization; Lasserre relaxation; Semidefinite program; Finite convergence; 90C22; 90C47; 49K35; 65K05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-025-00657-0

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