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<p>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.</p>

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All saddle points for polynomial optimization

  • 期刊论文

  • 发表日期：

    2025年2月7日

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  • Anwa Zhou,
  • Shiqian Yin,
  • Jinyan Fan

摘要

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.