Metric regularity of epigraphical multivalued mappings and applications to vector optimization

In this work we combine in a meaningful way two techniques of variational analysis and nonsmooth optimization. On one hand, we use the error bound approach to study the metric regularity of some special types of multifunctions and, on the other hand, we exploit the incompatibility between the metric regularity and the Pareto minimality. This method allows us to present some $varepsilon $ -Fermat rules for set-valued optimization problem in the setting of general Banach spaces. Our results are comparable to several recent results in literature.