A general representation of delta-normal sets to sublevels of convex functions

TL;DR

This paper characterizes the delta-normal cone to intersections of sublevel sets of convex functions using subdifferentials, providing a unified framework applicable in locally convex spaces with specific restrictions.

Contribution

It introduces a general representation of delta-normal sets for sublevels of convex functions, utilizing epsilon- and exact subdifferentials with calculus rules, extending existing theories.

Findings

Characterization of delta-normal cones via subdifferentials.

Use of epsilon-calculus rules for sup/max functions.

Framework applicable in locally convex spaces with certain restrictions.

Abstract

The (delta-) normal cone to an arbitrary intersection of sublevel sets of proper, lower semicontinuous, and convex functions is characterized, using either epsilon-subdifferentials at the nominal point or exact subdifferentials at nearby points. Our tools include (epsilon-) calculus rules for sup/max functions. The framework of this work is that of a locally convex space, however, formulas using exact subdifferentials require some restriction either on the space (e.g. Banach), or on the function (e.g. epi-pointed).