Lower Envelopes and Steepest Descent Directions in Vector Optimization

TL;DR

This paper characterizes the continuity of lower envelopes in infinite-dimensional spaces using c-regularity and applies this to vector optimization, identifying conditions for continuous steepest descent directions.

Contribution

It provides a complete characterization of lower envelope continuity via c-regularity and introduces conditions for continuous steepest descent directions in vector optimization.

Findings

Continuity of lower envelopes characterized by c-regularity.

Conditions for continuous steepest descent directions in vector optimization.

Application to smooth functions in infinite-dimensional spaces.

Abstract

The purpose of the paper is to give a complete characterization of the continuity of lower envelopes in the infinite dimensional spaces in terms of the notion of c-regularity. As an application we introduce a variational unconstrained vector optimization problem for smooth functions and characterize when the variational steepest descent directions are continuous in terms of the generating sets which are considered.