Steepest geometric descent for regularized quasiconvex functions

TL;DR

This paper proves the existence of steepest descent curves for regular quasiconvex functions and introduces a max-convolution regularization technique to extend these results to more general functions.

Contribution

It establishes the existence of steepest descent curves for regular quasiconvex functions and develops a max-convolution method for regularizing broader classes of quasiconvex functions.

Findings

Existence of steepest descent curves for almost every point in regular quasiconvex functions.

Reversibility of the sweeping process flow induced by sublevel sets.

Max-convolution regularization extends results to more general quasiconvex functions.

Abstract

We establish existence of steepest descent curves emanating from almost every point of a regular locally Lipschitz quasiconvex functions, where regularity means that the sweeping process flow induced by the sublevel sets is reversible. We then use max-convolution to regularize general quasiconvex functions and obtain a result of the same nature in a more general setting.