Mathematical programs with complementarity constraints in Banach spaces (Q493055)

The author extends optimality assertions for finite-dimensional optimization problems with complementarity constraints of the form \[ \text{Minimize}\quad f(x)\text{ s.t. }G(x)\geq 0,\;H(x)\geq 0,\;G(x)^\top H(x)= 0 \] (here \(f: \mathbb{R}^n\to\mathbb{R}\) and \(G,H: \mathbb{R}^n\to\mathbb{R}^m\) are smooth functions) to more general optimization problems with conical complementary conditions in infinite-dimensional spaces of the form \[ \text{Minimize }f(x)\text{ s.t. }g(x)\in C,\;G(x)\in K,\;H(x)\in K^\circ,\;\langle G(x), H(x)\rangle= 0. \] Also here \(f: X\to \mathbb{R}\), \(g: X\to Y\), \(G: X\to Z\), \(H: X\to Z^\ast\) are smooth functions, but \(X\), \(Y\) and \(Z\) are real Banach spaces, where \(Z\) is assumed to be reflexive, \(C\subset Y\) is a closed convex set and \(K\subset Z\) is a closed convex cone with dual cone \(K^\circ\subset Z^\ast\). It is known that for such problems the classical constraint qualifications fail in general, hence the Karush-Kuhn-Tucker conditions are not necessary for optimality. Based on different auxiliary problems (which eliminate the condition \(\langle G(x),H(x)\rangle= 0\)) the author introduces the notion of strong stationarity and provides a suitable constraint qualification such that a necessary optimality assertion can be formulated. If the cone \(K\) is even polyhedral, then it is shown that strong stationarity is equivalent to a known stationarity notion. In the last part of the paper, the author gives some remarks on the case where \(K\) is not a cone and provides two examples which illustrate the results.