Normal cones to sublevel sets of convex and quasi-convex supremum functions

TL;DR

This paper characterizes the normal cones to sublevel sets of supremum functions, providing explicit formulas in convex and quasi-convex cases, which are used to derive optimality conditions for complex optimization problems.

Contribution

It offers new explicit formulas for normal cones to sublevel sets of supremum functions in convex and quasi-convex settings, enhancing understanding of their geometric structure.

Findings

Explicit formulas for normal cones in convex case using subdifferentials.

Formulas for quasi-convex case involving Fréchet subdifferentials at nearby points.

Application of results to derive optimality conditions in infinite-dimensional optimization.

Abstract

We provide sharp and explicit characterizations of the normal cone to sublevel sets of suprema of arbitrary functions, expressed exclusively in terms of subdifferentials of the data functions. In the convex case, the resulting formulas involve the approximate subdifferential of the individual data functions at the nominal point. In contrast, the quasi-convex framework requires the use of the Fr\'echet subdifferential of these data functions but evaluated at nearby points. These results are applied to derive optimality conditions for infinite convex and quasi-convex optimization problems.