DUALITY THEOREM AND VECTOR SADDLE POINT THEOREM FOR ROBUST MULTIOBJECTIVE OPTIMIZATION PROBLEMS

Abstract

Abstract. In this paper, Mond-Weir type duality results for a uncertainmultiobjective robust optimization problem are given under generalizedinvexity assumptions. Also, weak vector saddle-point theorems are ob-tained under convexity assumptions. 1. IntroductionConsider an uncertain multiobjective robust optimization problem:(MRP) minimize (f 1 (x),...,f l (x))subject to g j (x,v j ) <= 0, ∀v j ∈ V j , j = 1,...,m,where v i is an uncertain parameter and v i ∈ V i for some convex compact setV i in R q , f i : R n → R, i = 1,...,l and g j : R n × R q → R, j = 1,...,m arecontinuously differentiable.When l = 1, (MRP) becomes an uncertain optimization problem, which hasbeen intensively studied in ([4]-[5], [6]), associates with the uncertain program(UP) its robust counterpart [1],(RP) inf x∈R n {f(x) : g i (x,v i ) ≤ 0, ∀v i ∈ V i , i = 1,...,m},where the uncertain constraints are enforced for every possible value of theparameters within their prescribed uncertainty sets V i , i = 1,...,m. Recently,Jeyakumar, Li and Lee [7] established a robust duality theory for generalizedconvex programming problems in the face of data uncertainty. Furthermore,Kim [8] extended results of Jeyakumar, Li and Lee [7] for a uncertain multi-objective robust optimization problem. In this paper, Mond-Weir type dualityresults for a uncertain multiobjective robust optimization problem are given