Weak Tangent Cones and Optimization in a Banach Space

Abstract

A general notion of a $\tau $-tangent cone is introduced and developed for optimization purposes. This includes as special cases both the weak and strong tangent cones that appear in the literature. First order conditions with and without constraint qualification are examined and particular examples are provided to demonstrate that these conditions properly subsume those previously in the literature. Emphasis is placed on weak Kuhn–Tucker sufficiency conditions.