Characterizing weak solutions for vector optimization problems

Abstract

This paper provides characterizations of the weak solutions of optimization problems where a given vector function $F,$ from a decision space $X$ to an objective space $Y$, is "minimized" on the set of elements $x\in C$ (where $C\subset X$ is a given nonempty constraint set), satisfying $G\left( x\right) \leqq_{S}0_{Z},$ where $G$ is another given vector function from $X $ to a constraint space $Z$ with positive cone $S$. The three spaces $X,Y,$ and $Z$ are locally convex Hausdorff topological vector spaces, with $Y$ and $Z$ partially ordered by two convex cones $K$ and $S,$ respectively, and enlarged with a greatest and a smallest element. In order to get suitable versions of the Farkas lemma allowing to obtain optimality conditions expressed in terms of the data, the triplet $\left( F,G,C\right) ,$ we use non-asymptotic representations of the $K-$epigraph of the conjugate function of $F+I_{A},$ where $I_{A}$ denotes the indicator function of the feasible set $A,$ that is, the function associating the zero vector of $Y$ to any element of $A$ and the greatest element of $Y$ to any element of $X\diagdown A.$