Singularities of Fitzpatrick and convex functions

TL;DR

This paper investigates the structure of singularities in Fitzpatrick functions and projections on S-monotone sets within pseudo-Euclidean spaces, revealing they are covered by countable surfaces with specific normal vector properties.

Contribution

It extends the understanding of singularities of convex and Fitzpatrick functions by characterizing their covering surfaces and the restrictions on their normal vectors in pseudo-Euclidean spaces.

Findings

Singularities are covered by countable c-c surfaces with positive S-normal vectors.

Normal vectors to these surfaces are confined to the cone generated by differences in the closure of the gradient's range.

The results generalize previous work on convex function singularities to pseudo-Euclidean settings.

Abstract

In a pseudo-Euclidean space with scalar product $S(\cdot, \cdot)$, we show that the singularities of projections on $S$-monotone sets and of the associated Fitzpatrick functions are covered by countable $c-c$ surfaces having positive normal vectors with respect to the $S$-product. By Zaj\'{\i}\v{c}ek [24], the singularities of a convex function $f$ can be covered by a countable collection of $c-c$ surfaces. We show that the normal vectors to these surfaces are restricted to the cone generated by $F-F$, where $F:=\text{cl range } \nabla f$, the closure of the range of the gradient of $f$.