## Complementary problems with polynomial data

Given polynomial maps $f, g \colon \mathbb{R}^n \to \mathbb{R}^n,$ we consider the {\em polynomial complementary problem} of finding a vector $x \in \mathbb{R}^n$ such that \begin{equation*} f(x) \ \ge \ 0, \quad g(x) \ \ge \ 0, \quad \textrm{ and } \quad \langle f(x), g(x) \rangle \ = \ 0. \end{equation*} In this paper, we … Read more

## Generic properties for semialgebraic programs

In this paper we study genericity for the following parameterized class of nonlinear programs: \begin{eqnarray*} \textrm{minimize } f_u(x) := f(x) – \langle u, x \rangle \quad \textrm{subject to } \quad x \in S, \end{eqnarray*} where $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ is a polynomial function and $S \subset \mathbb{R}^n$ is a closed semialgebraic set, which is … Read more

## Stability and genericity for semi-algebraic compact programs

In this paper we consider the class of polynomial optimization problems with inequality and equality constraints, in which every problem of the class is obtained by perturbations of the objective function, while the constraint functions are kept fixed. Under certain assumptions, we establish some stability properties (e.g., strong H\”older stability with explicitly determined exponents, semicontinuity, … Read more

## How to Solve a Semi-infinite Optimization Problem

After an introduction to main ideas of semi-infinite optimization, this article surveys recent developments in theory and numerical methods for standard and generalized semi-infinite optimization problems. Particular attention is paid to connections with mathematical programs with complementarity constraints, lower level Wolfe duality, semi-smooth approaches, as well as branch and bound techniques in adaptive convexification procedures. … Read more