Do Sang Kim

Vector variational inequality as a tool for studying vector optimization problems

The paper aims to show that a Vector Variational Inequality can be an useful tool for studying a Vector Optimization Problem.

Generalized vector variational inequality and fuzzy extension

Abstract A generalized vector variational inequality (GVVI) is considered. We establish the existence theorem for (GVVI) under assumptions of C -pseudomonotonicity and V -hemicontinuity. From our existence theorem, we obtain the fuzzy extension of a result of Chen and Yang.

Optimality and Duality for Invex Nonsmooth Multiobjective programming problems

We consider nonsmooth multiobjective programming problems with inequality and equality constraints involving locally Lipschitz functions. Several sufficient optimality conditions under various (generalized) invexity assumptions and certain regularity conditions are presented. In addition, we introduce a Wolfe-type dual and Mond–Weir-type dual and establish duality relations under various (generalized) invexity and regularity conditions.

Generalized vector variational inequality

A generalized vector variational inequality (GVVI) is considered. We establish the existence theorem for solutions of (GVVI) with H-convexity assumption. Our existence theorem extends that of Chen [1, Theorem 3.1] to the set-valued case.

Infine Functions, Nonsmooth Alternative Theorems and Vector Optimization Problems

In this paper we introduce a new notion of infine nonsmooth functions and give several characterizations of infineness property. We prove alternative theorems with mixed constraints (i.e., inequality and equality constraints) being described by invex-infine nonsmooth functions. We establish a necessary and sufficient condition for a solution of a vector optimization problem involving mixed constraints to be a properly efficient solution.

Nonsmooth Semi-infinite Multiobjective Optimization Problems

We apply some advanced tools of variational analysis and generalized differentiation to establish necessary conditions for (weakly) efficient solutions of a nonsmooth semi-infinite multiobjective optimization problem (SIMOP for brevity). Sufficient conditions for (weakly) efficient solutions of a SIMOP are also provided by means of introducing the concepts of (strictly) generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. In addition, we propose types of Wolfe and Mond–Weir dual problems for SIMOPs, and explore weak and strong duality relations under assumptions of (strictly) generalized convexity. Examples are also designed to analyze and illustrate the obtained results.

Non-differentiable symmetric duality for multiobjective programming with cone constraints

Abstract Two pairs of non-differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones, which are Wolfe type and Mond–Weir type, are considered. On the basis of weak efficiency with respect to a convex cone, we obtain symmetric duality results for the two pairs of problems under cone-invexity and cone-pseudoinvexity assumptions on the involved functions. Our results extend the results in Khurana [S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research 165 (2005) 592–597] to the non-differentiable multiobjective symmetric dual problem.

Generalized Vector Variational Inequality and its duality for set-valued maps

Abstract In this paper, we consider a Generalized Vector Variational Inequality (GVVI) for set-valued maps, give its dual form (DVVI), and prove an equivalence between (GVVI) and (DVVI).

Second-order symmetric and self duality in multiobjective programming☆

Abstract We suggest the second-order symmetric and self dual programs in multiobjective nonlinear programming. For these second-order symmetric dual programs, we prove the weak, strong, and converse duality theorems under convexity and concavity conditions. Also, we prove the self duality theorem for these second-order self dual programs and illustrate its example.

Duality for multiobjective programming involving n-set functions

We formulate the Wolfe, Mond-Weir and generalized Mond-Weir types of dual problems for a multiobjective program involving vector valued n-set functions. By using the concept of efficiency (Pareto optimum) we prove the weak and strong duality theorems for a multiobjective program under generalized ρ-convexity assumptions.