On types of KKT points in polynomial optimization

Abstract

Let $f$ be a real polynomial function with $n$ variables and $S$ be a basic closed semialgebraic set in ${\mathbb{R}}^n$. In this paper, we are interested in the problem of identifying the type (local minimizer, maximizer or not extremum point) of a given isolated KKT point $x^{\ast }$ of $f$ over $S.$ To this end, we investigate some properties of the tangency variety of $f$ on $S$ at $x^{\ast },$ by which we introduce the definition of faithful radius of $f$ over $S$ at $x^{\ast }.$ Then, we show that the type of $x^{\ast }$ can be determined by the global extrema of $f$ over the intersection of $S$ and the Euclidean ball centered at $x^{\ast }$ with a faithful radius. Finally, we propose an algorithm involving algebraic computations to compute a faithful radius of $x^{\ast }$ and determine its type.

29 pages