论文标题
多项式优化中的KKT点类型
On types of KKT points in polynomial optimization
论文作者
论文摘要
令$ f $是具有$ n $变量的真实多项式函数，$ s $是$ \ bbb {r}^n $的基本闭合半gebraic设置。在本文中，我们对确定给定的孤立的kkt点的类型（本地最小化，最大化器或不是极值点）感兴趣。 $ x^*$可以由$ s $的交叉点的全球极值和以忠实半径为$ x^*$的交叉点和欧几里得球确定。最后，我们提出了一种涉及代数计算的算法，以计算$ x^*$的忠实半径并确定其类型。
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.
