Duality In Multiobjective Programming Research Articles

A sublinear functional based approximated equivalence to optimality and duality for multiobjective programming in complex spaces

In this paper, we introduce a new approximation approach for a class of multiobjective programming problems in complex spaces and their duals. Using a sublinear functional, an approximated problem is constructed at a given feasible solution of the original problem. The equivalence between the solution of the considered multiobjective complex programming problem and its approximated problem is established by deriving the sufficiency theorem under the F-convexity assumption. An example showing this equivalence is also presented. Further, corresponding to the approximated problem, the Mond–Weir and Wolfe-type approximated dual problems are formulated. The duality results for the Mond–Weir and Wolfe-type dual problems are derived using the duality results established for the approximated Mond–Weir and Wolfe-type dual problems, respectively. The importance of the optimality conditions and proposed approximation approach is highlighted in a blind equalization problem in a communication system.

Modified converse duality for multiobjective higher order Wolfe type dual program with cone constraints

In this paper, we obtain a converse duality theorem for higher order Wolfe type multiobjective programming with cone constraints under appropriate assumptions. This fills some gaps in the work of Kim et al. [Kim, D.S., Kang, H.S., Lee, Y.J., Seo, Y.Y., Higher order duality in multiobjective programming with cone constraints, Optimization, 59(1), 29?43, (2010)].

Higher-order symmetric duality for multiobjective programming with cone constraints

In this work, a pair of higher-order symmetric dual multiobjective optimization problems is formulated. Weak, strong and converse duality theorems are established under suitable assumptions. Some examples are also given to illustrate our main results. Furthermore, certain deficiencies in the formulations and the proof of the work of Kassem [Applied Mathematics and Computation, 209 (2009), 405-409] are pointed out.

Optimality and duality in multiobjective programming involving support functions

In this paper a vector optimization problem (VOP) is considered where each component of objective and constraint function involves a term containing support function of a compact convex set. Weak and strong Kuhn−Tucker necessary optimality conditions for the problem are obtained under suitable constraint qualifications. Necessary and sufficient conditions are proved for a critical point to be a weak efficient or an efficient solution of the problem (VOP) assuming that the functions belong to different classes of pseudoinvex functions. Two Mond Weir type dual problems are considered for (VOP) and duality results are established.

Optimality and duality in multiobjective programming involving higher order semilocally strong convexity

In this paper, we propose a generalisation of convexity, namely higher order semilocally strong convexity for a nonlinear multiobjective programming problem, where the function involved are semidifferentiable. The generalised Karush-Kuhn-Tucker necessary and sufficient optimality conditions are derived. Moreover, a general Mond-Weir type dual problem is presented for nonlinear multiobjective programming problem involving higher order semilocally strong convexity and usual duality theorems are discussed.

Optimality and duality in multiobjective programming involving higher order semilocally strong convexity

In this paper, we propose a generalisation of convexity, namely higher order semilocally strong convexity for a nonlinear multiobjective programming problem, where the function involved are semidifferentiable. The generalised Karush-Kuhn-Tucker necessary and sufficient optimality conditions are derived. Moreover, a general Mond-Weir type dual problem is presented for nonlinear multiobjective programming problem involving higher order semilocally strong convexity and usual duality theorems are discussed.

Second order duality for multiobjective programming with cone constraints

We focus on second order duality for a class of multiobjective programming problem subject to cone constraints. Four types of second order duality models are formulated. Weak and strong duality theorems are established in terms of the generalized convexity, respectively. Converse duality theorems, essential parts of duality theory, are presented under appropriate assumptions. Moreover, some deficiencies in the work of Ahmad and Agarwal (2010) are discussed.

Sufficiency and Duality for Multiobjective Programming under New Invexity

A class of multiobjective programming problems including inequality constraints is considered. To this aim, some new concepts of generalizedF,P-type I andF,P-type II functions are introduced in the differentiable assumption by using the sublinear functionF. These new functions are used to establish and prove the sufficient optimality conditions for weak efficiency or efficiency of the multiobjective programming problems. Moreover, two kinds of dual models are formulated. The weak dual, strong dual, and strict converse dual results are obtained under the aforesaid functions.

Mond-Weir Duality for Multiobjective Programming with Invexity

Mond-Weir Duality for Multiobjective Programming with Invexity

Non-differentiable multiobjective programming under generalised functions

In this paper, we consider a nonlinear multiobjective programming problem where functions involved are non-differentiable. The use of directional derivative in association with a hypothesis of an invex kind on the set has been of much interest in the recent past. Recently under preinvexity assumptions on fi′(xo;ηi(x,xo)) and g′j (xo;θj(x,xo)), necessary and sufficient optimality conditions for a non-smooth multiobjective optimisation problem under generalised class of dI-V-type I functions have been derived. Working in this direction, we introduce a new class of (φ, dI)-V-type I functions and illustrate through various non-trivial examples that this class is non-empty and extends some known classes introduced in the literature. Also, we obtain sufficient optimality conditions to enable a feasible solution of the primal problem to be its weak efficient/efficient solution. Then through an example, we illustrate the relevance of the sufficient optimality theorem obtained to get the efficient solution of the problem. Further, we formulate Wolfe type and Mond-Weir type multiobjective dual programs and establish various duality theorems under this newly introduced class of functions.

Second Order Duality in Multiobjective Programming With Generalized Convexity

The purpose of this paper is concerned with a class of multi objective programming problems in which the objective functions and constrained functions are twice differentiable and containing the support functions of a compact convex set. For such mathematic programming, to deal with its dual problem, first, the concept of second order ( , ) B p r  invexity is introduced. Further, the Mangasarian type second order dual model associated with the multi objective problem are formulated. Several weak, strong and strict converse dual theorems are established and proved by utilizing the new generalized convexity. The results extend and improve the corresponding results in the literature.

Second-Order Nondifferentiable Multiobjective Mixed Symmetric Dual Programs Over Cones

In this paper, a pair of second-order mixed symmetric nondifferentiable multiobjective dual programs over arbitrary cones where each of the objective functions contains a pair of support functions is considered. Further, weak, strong and converse duality theorems are established under the assumptions of second-order K- (F,ρ)-convexity. Moreover, we have illustrated a numerical example which is second-order K- (F,ρ)-convex but not second-order F-convex. Several special cases are also obtained.

Wolfe Type Second Order Nondifferentiable Symmetric Duality in Multiobjective Programming over Cone with Generalized (K, F)-Convexity

A new class of second order (K, F) pseudoconvex function is introduced with example. A pair of Wolfe type second order nondifferentiable symmetric dual programs over arbitrary cones with square root term is formulated. The duality results are established under second order (K, F) pseudoconvexity assumption. Also a Wolfe type second order minimax mixed integer programming problem is formulated and the symmetric duality results are established under second order (K, F) pseudoconvexity assumption.

Duality and Saddle Point for Multiobjective Semi-infinite Programming

This paper is devoted to study the mixed dual models for a class of non-smooth multiobjective semi-infinite programming problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. Weak duality conclusions are derived and proved for mixed type multiobjective dual programs, using the generalized uniform convexity on the functions involved. Some previous duality results for differentiable semi-infinite programming problems turn out to be special cases for the results described in the paper. Furthermore, the vector saddle point theory is discussed for the semi-infinite programming problems. The results extend and improve the corresponding results in the literature.

On Nondifferentiable Higher-Order Symmetric Duality in Multiobjective Programming Involving Cones

In this paper, we point out some deficiencies in a recent paper (Lee and Kim in J. Nonlinear Convex Anal. 13:599–614, 2012), and we establish strong duality and converse duality theorems for two types of nondifferentiable higher-order symmetric duals multiobjective programming involving cones.

Sufficiency and duality for nonsmooth multiobjective programming problems involving generalized univex functions

In this paper, nonsmooth univex, nonsmooth quasiunivex, and nonsmooth pseudounivex functions are introduced. By utilizing these new concepts, sufficient optimality conditions for a weakly efficient solution of the nonsmooth multiobjective programming problem are established. Weak and strong duality theorems are also derived for Mond-Weir type multiobjective dual programs.

Higher-order Mond-Weir converse duality in multiobjective programming involving cones

In this work, we established a converse duality theorem for higher-order Mond-Weir type multiobjective programming involving cones. This fills some gap in recently work of Kim et al. [Kim D S, Kang H S, Lee Y J, et al. Higher order duality in multiobjective programming with cone constraints. Optimization, 2010, 59: 29–43].

Higher-Order Duality for Multiobjective Programming Problems Involving (F, α, ρ, d)-V-Type I Functions

A class of functions called higher-order (F, α, ρ, d)-V-type I functions and their generalizations is introduced. Using the assumptions on the functions involved, weak, strong and strict converse duality theorems are established for higher-order Wolfe and Mond-Weir type multiobjective dual programs in order to relate the efficient solutions of primal and dual problems.

Optimality and Duality for Non-smooth Multiple Objective Semi-infinite Programming

The purpose of this paper is to consider a class of non-smooth multiobjective semi-infinite programming problem. Based on the concepts of local cone approximation, directional derivative and subdifferential, a new generalization of convexity, namely generalized uniform convexity, is defined for this problem. For such semi-infinite programming problem, several sufficient optimality conditions are established and proved by utilizing the above defined new classes of functions. The results extend and improve the corresponding results in the literature. Subsequently, these optimality conditions are utilized as a basis for formulating dual problems. Weak, strong and reverse duality theorems are also derived for Mond-Weir type multiobjective dual programs, using generalized invexity on the functions involved. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases for the results described in the paper.

Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs

In this paper, we establish a strong duality theorem for Mond-Weir type multiobjective higher order nondifferentiable symmetric dual programs. Our works correct some deficiencies in recent papers [higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 290(2004)423-435] and [A note on higher-order nondifferentiable symmetric duality in multiobjective programming, Appl. Math. Letters 24(2011) 1308-1311].