Curves of Descent

TL;DR

This paper provides a new existence proof for curves of near-maximal slope in nonsmooth optimization and characterizes these curves for semi-algebraic functions as solutions to subgradient dynamical systems.

Contribution

It introduces a transparent existence proof for curves of near-maximal slope and links them to subgradient dynamical systems for semi-algebraic functions.

Findings

Existence of curves of near-maximal slope established.

Characterization of these curves as solutions to subgradient systems for semi-algebraic functions.

Enhances understanding of steepest descent in nonsmooth optimization.

Abstract

Steepest descent is central in variational mathematics. We present a new transparent existence proof for curves of near-maximal slope --- an influential notion of steepest descent in a nonsmooth setting. We moreover show that for semi-algebraic functions --- prototypical nonpathological functions in nonsmooth optimization --- such curves are precisely the solutions of subgradient dynamical systems.