SIP: critical value functions have finite modulus of non-convexity

Abstract

We consider semi-infinite programming problems ${{\rm SIP}(z)}$ depending on a finite dimensional parameter ${z \in \mathbb{R}^p}$. Provided that ${\bar{x}}$ is a strongly stable stationary point of ${{\rm SIP}(\bar{z})}$, there exists a locally unique and continuous stationary point mapping ${z \mapsto x(z)}$. This defines the local critical value function ${\varphi(z) := f(x(z); z)}$, where ${x \mapsto f(x; z)}$ denotes the objective function of ${{\rm SIP}(z)}$ 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 ${\varphi}$.