Duality In Multiobjective Programming Research Articles

Second order symmetric duality in multiobjective programming involving cones

An attempt is made to remove certain omissions and inconsistencies in the recent work of Mishra and Lai (European J. Oper. Res., 178:20–26, 2007).

A note on higher-order nondifferentiable symmetric duality in multiobjective programming

In this work, we establish a strong duality theorem for Mond–Weir type multiobjective higher-order nondifferentiable symmetric dual programs. This fills some gaps in the work of Chen [X. Chen, Higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 290 (2004) 423–435].

Higher order duality in multiobjective programming with cone constraints

We consider the nonlinear multiobjective programming problems involving cone constraints. For this program, we construct higher order dual problems and establish weak, strong and converse duality theorems for an efficient solution by using higher order generalized invexity conditions due to Zhang (Higher order convexity and duality in multiobjective programming, in “Progress in Optimization”, Contributions from Australasia, Applied Optimization, Vol. 30, eds. A. Eberhard, R. Hill, D. Ralph and B.M. Glover, Kluwer Academic Publishers, Dordrecht, 1999, 101–116). As special cases of our duality relations, we give some known duality results.

Mond–Weir type second-order symmetric duality in multiobjective programming over cones

A pair of Mond–Weir type second-order symmetric dual multiobjective programs over arbitrary cones is formulated. Weak, strong and converse duality theorems are established under pseudoinvexity/ K – F -convexity assumptions.

On some symmetric dual models in multiobjective programming

Certain omissions have been pointed out in some papers on symmetric duality in multiobjective programming. Corrective measures have also been discussed.

Nondifferentiable higher order duality in multiobjective programming involving cones

We introduce the nondifferentiable multiobjective problem involving cone constraints, where every component of the objective function contains a term involving the support function of a compact convex set. For this problem, Wolfe and Mond–Weir type duals are proposed. We establish weak, strong duality theorems for an efficient solution under suitable higher order generalized invexity conditions. As special cases of our duality relations, we give some known duality 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.

Second Order Duality in Multiobjective Programming

A nonlinear multiobjective programming problem is considered. Weak, strong and strict converse duality theorems are established under generalized second order (F, α, ρ, d)-convexity for second order Mangasarian type and general Mond-Weir type vector duals.

Optimality conditions and duality in multiobjective programming with invexity

(', ?)-invexity has recently been introduced with the intent of generalizing invex functions in mathematical programming. Using such conditions we obtain new sufficiency results for optimality in multiobjective programming and extend some classical duality properties.

Higher-order symmetric duality in multiobjective programming with invexity

In this paper, a pair of higher order symmetric dual models for multiobjective nonlinear programming is introduced. The weak, strong and converse duality theorems are proven for the formulated higher order symmetric dual programs under invexity conditions.

Second-Order Duality in Multiobjective Programming Involving (F, α, ρ, d)-V-Type I Functions

In this paper, a new class of second-order (F, α, ρ, d)-V-type I functions is introduced that generalizes the notion of (F, α, ρ, θ)-V-convex functions introduced by Zalmai (Computers Math. Appl. 2002; 43:1489–1520) and (F, α, ρ, p, d)-type I functions defined by Hachimi and Aghezzaf (Numer. Funct. Anal. Optim. 2004; 25:725–736). Based on these functions, weak, strong, and strict converse duality theorems are derived for Wolfe and Mond–Weir type multiobjective dual programs in order to relate the efficient and weak efficient solutions of primal and dual problems.

Sufficiency and Duality in Multiobjective Programming under Generalized Type I Functions

In this paper, new classes of generalized (F,α,ρ,d)-V-type I functions are introduced for differentiable multiobjective programming problems. Based upon these generalized convex functions, sufficient optimality conditions are established. Weak, strong and strict converse duality theorems are also derived for Wolfe and Mond-Weir type multiobjective dual programs.

Second order symmetric duality in multiobjective programming involving generalized cone-invex functions

In this paper, cone-second order pseudo-invex and strongly cone-second order pseudo-invex functions are defined. A pair of Mond–Weir type second order symmetric dual multiobjective programs is formulated over arbitrary cones. Weak, strong and converse duality theorems are established under aforesaid generalized invexity assumptions. A second self-duality theorem is also given by assuming the functions involved to be skew-symmetric.

Symmetric and self duality in vector optimization problem

In this paper, we present the first-order symmetric and self duality programs in multiobjective nonlinear programming problems. For these first-order symmetric and dual programs, we introduce the weak, strong, and converse duality theorems under convexity and concavity conditions, where the pair of the symmetric dual multiobjective nonlinear programming problems in this paper different which in [D.S. Kim, Y.B. Yun, H. Kuk, Second-order symmetric and self duality in multiobjective programming. Appl. Math. Lett. 10(2) (1997) 17–22]. Also, we prove the self duality theorem for these first-order self dual programs and illustrate its example.

Optimality Criteria and Duality in Multiobjective Programming Involving Nonsmooth Invex Functions

In this paper a generalization of invexity is considered in a general form, by means of the concept of K-directional derivative. Then in the case of nonlinear multiobjective programming problems where the functions involved are nondifferentiable, we established sufficient optimality conditions without any convexity assumption of the K-directional derivative. Then we obtained some duality results.

Sufficiency and duality in nonsmooth multiobjective optimization involving generalized ( F, α, π, d)-Type I functions

In this paper, new classes of generalized ( F, α, π, d)-Type I functions are introduced for a nonsmooth multiobjective programming problem. Based upon these generalized functions, sufficient optimality conditions are established. Weak, strong, and strict converse duality theorems are also derived for Wolfe and Mond-Weir type multiobjective dual programs in order to relate the efficient and weak efficient solutions of primal and dual problems.

Saddle point criteria and duality in multiobjective programming via an η-approximation method

AbstractIn this paper, Antczak's η-approximation approach is used to prove the equivalence between optima of multiobjective programming problems and the η-saddle points of the associated η-approximated vector optimisation problems. We introduce an η-Lagrange function for a constructed η-approximated vector optimisation problem and present some modified η-saddle point results. Furthermore, we construct an η-approximated Mond-Weir dual problem associated with the original dual problem of the considered multiobjective programming problem. Using duality theorems between η-approximation vector optimisation problems and their duals (that is, an η-approximated dual problem), various duality theorems are established for the original multiobjective programming problem and its original Mond-Weir dual problem.

Second order ( F, α, ρ, d)-convexity and duality in multiobjective programming

A class of second order ( F, α, ρ, d)-convex functions and their generalizations is introduced. Using the assumptions on the functions involved, weak, strong and strict converse duality theorems are established for a second order Mond–Weir type multiobjective dual.

Symmetric duality in multiobjective programming involving generalized cone-invex functions

In this paper, cone-pseudoinvex and strongly cone-pseudoinvex functions are defined. A pair of Mond–Weir type symmetric dual multiobjective programs is formulated over arbitrary cones. Weak duality, strong duality and converse duality theorems are established using the above-defined functions. A self-duality theorem is also given by assuming the functions involved to be skew-symmetric.

Nondifferentiable second order symmetric duality in multiobjective programming

A pair of Mond–Weir type nondifferentiable multiobjective second order symmetric dual programs is formulated and symmetric duality theorems are established under the assumptions of second order F-pseudoconvexity/F-pseudoconcavity.