Holder continuity of the steepest descent direction for multiobjective optimization

TL;DR

This paper investigates the continuity properties of the multiobjective steepest descent direction for smooth functions, establishing it as Holder continuous with exponent 1/2 and not Lipschitz continuous even for polynomial objectives.

Contribution

It provides a precise characterization of the Holder continuity of the steepest descent direction in multiobjective optimization, revealing its optimal exponent and limitations.

Findings

Steepest descent direction is Holder continuous with exponent 1/2.

The direction is not Lipschitz continuous for polynomial objectives.

Optimality of the Holder exponent is demonstrated.

Abstract

The aim of this manuscript is to characterize the continuity properties of the multiobjective steepest descent direction for smooth objective functions. We will show that this direction is Holder continuous with optimal exponent 1/2. In particular, this direction fails to be Lipschitz continuous even for polynomial objectives.