The reduction ansatz in absence of lower semi-continuity (Q2722332)

The article under review is a valuable contribution to a deep understanding in analysis and topology of parametric (standard) semi-infinite or generalized semi-infinite optimization.NEWLINENEWLINENEWLINEFor example, in the latter area of nonlinear optimization, by definition the objective function \(f\) has to minimized subject to a possibly infinite number of inequality constraints. This gives rise to consider the lower stage optimization problem of the form NEWLINE\[NEWLINEP(x):\min g(x,y)\quad\text{s.t. }y\in Y(x)(\subseteq\mathbb{R}^m)NEWLINE\]NEWLINE such that the feasible set of \(M\) on the original (upper) stage can be written as the level set \(M= \{x\in\mathbb{R}^n\mid \varphi(x)\geq 0\}\), where \(\varphi(x)= \inf_{y\in Y(x)} g(x,y)\). In the literature, the mapping \(Y(\cdot)\) is often supposed to be finitely constrained by equality and inequality constraints. In the special case of standard semi-infinite optimization, i.e., where \(Y=\text{const}\), there already exists a well-elaborate anaytical nondegeneracy condition (applying the implicit function theorem) for a strong problem reduction, locally representing \(M\) by only a finite number of inequalities: Reduction Ansatz. This, however, is guaranteed (by constraint qualifications) under some upper and lower semicontinuity behaviour of \(Y\). In the author's generalization of the Reduction Ansatz to the \(x\)-dependent case \(Y= Y(x)\), upper (but no lower) semi-continuity is assumed only and, furthermore, the structural assumption of \(Y(x)\) as being finitely constrained is dropped. Under this weak preassumption, local reduction is guaranteed in two results, called Partial Reduction Ansatz and Reduction Ansatz, which are implications based on the local uniqueness property of continuously depending local minimizers of the lower stage problem.NEWLINENEWLINENEWLINEThen, there are two applications presented. The first application is due to the case where \(Y(x)\) is finitely constrained, where the number of equations is \(m-1\). Here, special attention is paid to feasible boundary points (of \(M\)) of maximal active index type (mai-type) (being some kind of nondegeneracy, and the set of active \(y\) consisting of a singleton). Here, as being shown, \(M\) can be locally ``linearized'' (by a diffeomorphism). The second application takes places in parametric standard semi-infinite programming, where the generalized critical point theory of Jongen, Jonker and Twilt becomes partially extended.NEWLINENEWLINENEWLINEThis research is proved, examplified and organized clearly and with care.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00047].