One-parameter families of feasible sets in semi-infinite optimization
The object of this paper is to show how changes the global topological structure of one-parameter families of feasible sets \(M(t) \subseteq \mathbb{R}^n\) as the parameter \(t \in \mathbb{R} \) varies. The investigations are made for a quite natural class of constraints which describe \(M(t)\) with the following form \[ M^{(H,G,U,V)} (t) = \{ x \in \mathbb{R} ^n \mid h_i(x,t)=0, \;i\in I, \;G(x,y,t) \geq 0, \;y \in Y^{(U,V)}(t) \}, \] where \[ Y^{(U,V)} (t) = \{ y \in \mathbb{R} ^r \mid u_i(y,t)=0, \;i \in A, \;v_j(y,t) \geq 0, \;j \in B \} \] and \(h_i, G, u_i, v_j\) are \(p\)-times continuously differentiable functions. It is shown, that a change in the topological structure is closely related with the violation of some constraint qualification. The authors find out that there are five possible cases for a change in the topological structure of \(M(t)\).
scientific article; zbMATH DE number 1305617