Borwein-Preiss Vector Variational Principle

TL;DR

This paper extends the Borwein-Preiss vector variational principle to vector spaces, introducing new concepts of epsilon-minimality and strengthening previous metric space results.

Contribution

It generalizes the variational principle to vector settings and introduces novel epsilon-minimality concepts dependent on ordering cones and gauge functions.

Findings

Extended the Borwein-Preiss variational principle to vector spaces.

Introduced two new concepts of epsilon-minimality.

Strengthened previous metric space results.

Abstract

This article extends to the vector setting the results of our previous work Kruger et al. (2015) which refined and slightly strengthened the metric space version of the Borwein-Preiss variational principle due to Li and Shi, J. Math. Anal. Appl. 246(1), 308-319 (2000). We introduce and characterize two seemingly new natural concepts of epsilon-minimality, one of them dependent on the chosen element in the ordering cone and the fixed "gauge-type" function.