Characterizing weak solutions for vector optimization problems

TL;DR

This paper characterizes weak solutions for vector optimization problems in locally convex spaces, providing new optimality conditions and Farkas lemma variants using conjugate function representations.

Contribution

It introduces novel characterizations of weak solutions and develops Farkas lemma variants for vector optimization in topological vector spaces.

Findings

Provides conditions for weak solutions in vector optimization

Develops new Farkas lemma variants for optimality conditions

Uses conjugate function representations for analysis

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 $X\diagdown A.$