On types of KKT points in polynomial optimization

Feng Guo; Do Sang Kim; Liguo Jiao; Tien-Son Pham

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arXiv · 2004.12315

On types of KKT points in polynomial optimization

Feng Guo·Do Sang Kim·Liguo Jiao·Tien-Son Pham

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Abstract

Let $f$ be a real polynomial function with $n$ variables and $S$ be a basic closed semialgebraic set in $\Bbb{R}^n$. 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 $x^*$ of $f$ over $S.$ To this end, we investigate some properties of the tangency variety of $f$ on $S$ at $x^*,$ by which we introduce the definition of faithful radius of $f$ over $S$ at $x^*.$ Then, we show that the type of $x^*$ can be determined by the global extrema of $f$ over the intersection of $S$ and the Euclidean ball centered at $x^*$ with a faithful radius. Finally, we propose an algorithm involving algebraic computations to compute a faithful radius of $x^*$ and determine its type.

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Feng Guo, Do Sang Kim, Liguo Jiao, Tien-Son Pham. 2020-04-26. On types of KKT points in polynomial optimization. https://arxiv.org/abs/2004.12315

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