SIP: critical value functions have finite modulus of non-convexity
The authors consider semi-infinite programming problems \(SIP(z)\) depending on a finite dimensional parameter \(z \in \mathbb{R}^p\). In this work, the authors generalize the results obtained by \textit{H. Günzel} et al. [SIAM J. Optim. 16, No. 4, 1044--1053 (2006; Zbl 1131.90044)] to the case of critical value functions of parametric semi-infinite programs (SIP). As in [Zbl 1131.90044], the techniques used in order to deduce their results, strongly depend on the characterization of strong stability of stationary points. In the abstract it is said: ``Provided that \(\bar x\) is a strongly stable stationary point of \(SIP(\bar z)\), there exists a locally unique and continuous stationary point mapping \(z\rightarrow x(z)\). This defines the local critical value function \(\varphi(z)@>\text{def}>>{=}f(x(z);z)\), where \(x\rightarrow f(x;z)\) denotes the objective function of \(SIP(z)\) for a given parameter vector \(z \in \mathbb{R}^p\). We show that \(\varphi\) is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of \(\varphi\).''
scientific article; zbMATH DE number 6116931
scientific article