Characterizing weak solutions for vector optimization problems

2016/02/10 by Dinh, Nguyen, Goberna, Miguel A., Long, Dang H. +1
#49N15 #58E17 #90C29 #90C46 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1602.03367

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∈ C (where C⊂ X is a given nonempty constraint set), satisfying G( x) \leqq_S0_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 ( F,G,C) , 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.

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