On the restrictiveness of sequential normal compactness in function spaces

Patrick Mehlitz

Sequential normal compactness (SNC) is one of the fundamental properties in modern variational analysis. Its presence is necessary for the derivation of the extremal principle in Banach spaces and, therefore, calculus rules for the computation of generalized normals to set intersections or preimages of sets under transformations. While the SNC-property is inherent in finite-dimensional Banach spaces, its validity has to be checked carefully in the infinite-dimensional setting.

In this talk, we first summarize some theoretical results on the SNC-property. Afterwards, we discuss whether or not sets of the type
\{y\in \mathcal F(\Omega;\mathbb R^q)\,|\,y(\omega)\in C(\omega)\text{ f.a.a. }\omega\in\Omega\},

which frequently appear in the context of control- or state-constrained optimal control, possess the SNC-property. Here, $\mathcal F(\Omega;\mathbb R^q)$ represents a Banach space of vector-valued functions defined on a domain $\Omega\subset\mathbb R^d$ equipped with Lebesgue's measure, e.g.\ $L^p(\Omega;\mathbb R^q)$ or $W^{1,p}(\Omega;\mathbb R^q)$ for $p\in(1,\infty)$, and $C\colon\Omega\rightrightarrows\mathbb R^q$ is a measurable set-valued mapping. Finally, we comment on the consequences of our results for the optimization community. Exemplarily, we focus on complementarity-constrained optimization problems in function spaces.