Moon Hee Kim

Remarks on relations between vector variational inequality and vector optimization problem

In this paper, we study equivalent relations between vector variational inequalities for subdifferentials and nondifferentiable convex vector optimization problems. Futhermore, using the equivalent relations, we give existence theorems for solutions of convex vector optimization problems.

ROBUST DUALITY FOR GENERALIZED INVEX PROGRAMMING PROBLEMS

Abstract. In this paper we present a robust duality theory for gener-alized convex programming problems under data uncertainty. Recently,Jeyakumar, Li and Lee [Nonlinear Analysis 75 (2012), no. 3, 1362–1373]established a robust duality theory for generalized convex programmingproblems in the face of data uncertainty. Furthermore, we extend re-sults of Jeyakumar, Li and Lee for an uncertain multiobjective robustoptimization problem. 1. IntroductionConsider the standard nonlinear programming problem with inequality con-straints(P) inf x∈R n {f(x) : g i (x) <= 0, i = 1,...,m},where f : R n → Rand g i : R n → Rare continuously differentiable functions.The problem in the face of data uncertainty in the constraints can be capturedby the following nonlinear programming problem:(UP) inf x∈R n {f(x) : g i (x,v i ) <= 0, i = 1,...,m},where v i is an uncertain parameter and v i ∈ V i for some convex compact setV i in R q and g i : R n ×R q → Ris continuously differentiable. Robust optimiza-tion, which has emerged as a powerful deterministic approach for studyingmathematical programming under uncertainty ([4]-[5], [6]), associates with theuncertain program (UP) its robust counterpart [1],(RP) inf

On ε-optimality conditions for multiobjective fractional optimization problems

A multiobjective fractional optimization problem (MFP), which consists of more than two fractional objective functions with convex numerator functions and convex denominator functions, finitely many convex constraint functions, and a geometric constraint set, is considered. Using parametric approach, we transform the problem (MFP) into the non-fractional multiobjective convex optimization problem (NMCP) v with parametric v ∈ ℝ p , and then give the equivalent relation between (weakly) ε-efficient solution of (MFP) and (weakly) -efficient solution of . Using the equivalent relations, we obtain ε- optimality conditions for (weakly) ε- efficient solution for (MFP). Furthermore, we present examples illustrating the main results of this study.2000 Mathematics Subject Classification: 90C30, 90C46.

Relations between vector continuous-time program and vector variational-type inequality

In this paper, we establish relations between a solution of a vector continuous-time program and a solution of a vector variational-type inequality problem with functionals.

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

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

On linear vector program and vector matrix game equivalence

We define a vector matrix game, which is a vector version of the usual matrix game and consists of more than two skew symmetric matrices and vector ordering. Using vector optimization techniques, we characterize solutions for the vector matrix game. We establish equivalent relations between a linear vector optimization problem and its corresponding vector matrix game.

On vector matrix game and symmetric dual vector optimization problem

A vector matrix game with more than two skew symmetric matrices, which is an extension of the matrix game, is defined and the symmetric dual problem for a nonlinear vector optimization problem is considered. Using the Kakutani fixed point theorem, we prove an existence theorem for a vector matrix game. We establish equivalent relations between the symmetric dual problem and its related vector matrix game. Moreover, we give an example illustrating the equivalent relations.