A Moment-SOS Hierarchy for Robust Polynomial Matrix Inequality Optimization with SOS-Convexity

We study a class of polynomial optimization problems with a robust polynomial matrix inequality constraint for which the uncertainty set is defined also by a polynomial matrix inequality (including robust polynomial semidefinite programs as a special case). Under certain SOS-convexity assumptions, we construct a hierarchy of moment-SOS relaxations for this problem to obtain convergent upper … Read more

On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems

We consider the semi-infinite system of polynomial inequalities of the form \[ \mathbf{K}:=\{x\in\mathbb{R}^m\mid p(x,y)\ge 0,\ \ \forall y\in S\subseteq\mathbb{R}^n\}, \] where $p(X,Y)$ is a real polynomial in the variables $X$ and the parameters $Y$, the index set $S$ is a basic semialgebraic set in $\mathbb{R}^n$, $-p(X,y)$ is convex in $X$ for every $y\in S$. We … Read more

On types of degenerate critical points of real polynomial functions

In this paper, we consider the problem of identifying the type (local minimizer, maximizer or saddle point) of a given isolated real critical point $c$, which is degenerate, of a multivariate polynomial function $f$. To this end, we introduce the definition of faithful radius of $c$ by means of the curve of tangency of $f$. … Read more