Computing local minimizers in polynomial optimization under genericity conditions

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 $${{\,\mathrm{\mathbb {R}}\,}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mi>R</mml:mi> <mml:mspace/> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , 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.