An approximation of feasible sets in semi-infinite optimization.
The paper deals with the feasible set \(M:=\{x\in\mathbb{R}^{n}\mid h_{i}(x)=0,\;i\in I,\;G(x,y)\leq0,\) \(y\in Y\}\) of a semi-infinite optimization problem, where \(I\) is a finite set, \(Y\subset\mathbb{R}^{r}\) is a compact set and \(h_{i}\in C^{2}(\mathbb{R}^{n},\mathbb{R})\) for \(i\in I\) and \(G\in C^{2}(\mathbb{R}^{n}\times\mathbb{R}^{r},\mathbb{R}).\) Under the extended Mangasarian-Fromovitz constraint qualification condition, assumed to hold at every \(x\in M,\) it is shown that there exist an open subset \(U\) of \(\mathbb{R}^{r}\) with \(M\subset U\) and a unique continuous function \(x^{\ast }:U\rightarrow M\) with \(\left\| z-x^{\ast}(z)\right\| =d_{M}(z)\) (the distance from \(z\) to \(M\)) and \(z\in N(x^{\ast}(z))\) (the normal cone of \(M\) at \(x^{\ast}(z)\)) such that the restriction of \(d_{M}^{2}\) to \(U\) is continuously differentiable and \(Dd_{M}^{2}(z)=2(z-x^{\ast}(z))\) for \(z\in U.\)
extended Mangasarian-Fromovitz constraint qualification
scientific article; zbMATH DE number 1907716
scientific article