A Set-Valued Ekeland's Variational Principle in Vector Optimization

Abstract

This paper deals with Ekeland's variational principle for vector optimization problems. By using a set-valued metric, a set-valued perturbed map, and a cone-boundedness concept based on scalarization, we introduce an original approach to extending the well-known scalar Ekeland's principle to vector-valued maps. As a consequence of this approach, we obtain an Ekeland's variational principle that does not depend on any approximate efficiency notion. This result is related to other Ekeland's principles proved in the literature, and the finite-dimensional case is developed via an $\varepsilon$-efficiency notion that we introduced in [Math. Methods Oper. Res., 64 (2006), pp. 165–185; SIAM J. Optim., 17 (2006), pp. 688–710].