Higher-order Symmetric Duality Research Articles

A New Approach on Multiobjective Higher-Order Symmetric Duality Under Cone-Invexity

In this paper, we introduce a new pair of multiobjective higher-order symmetric dual models with cone constraints. Weak, strong and converse duality theorems for a such pair are discussed under higher-order cone-invex functions. Three non-trivial examples are presented to show the uniqueness of higher-order cone-invex function and existence of the multiobjective higher-order symmetric dual model. Several known results are obtained as special cases.

Higher-order symmetric duality in nondifferentiable multiobjective fractional programming problem over cone contraints

In this paper, we introduce the definition of higher-order K-(C, α, ρ, d)-convexity/pseudoconvexity over cone and discuss a nontrivial numerical examples for existing such type of functions. The purpose of the paper is to study higher order fractional symmetric duality over arbitrary cones for nondifferentiable Mond-Weir type programs under higher- order K-(C, α, ρ, d)-convexity/pseudoconvexity assumptions. Next, we prove appropriate duality relations under aforesaid assumptions.

Mixed Type Nondifferentiable Higher-Order Symmetric Duality over Cones

A new mixed type nondifferentiable higher-order symmetric dual programs over cones is formulated. As of now, in the literature, either Wolfe-type or Mond–Weir-type nondifferentiable symmetric duals have been studied. However, we present a unified dual model and discuss weak, strong, and converse duality theorems for such programs under higher-order F - convexity/higher-order F - pseudoconvexity. Self-duality is also discussed. Our dual programs and results generalize some dual formulations and results appeared in the literature. Two non-trivial examples are given to show the uniqueness of higher-order F - convex/higher-order F - pseudoconvex functions and existence of higher-order symmetric dual programs.

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.

Higher-order symmetric duality in nondifferentiable multiobjective optimization over cones

In this paper, a new pair of higher-order nondifferentiable multiobjective symmetric dual programs over arbitrary cones is formulated, where each of the objective functions contains a support function of a compact convex set. We identify a function lying exclusively in the class of higher-order K-?-convex and not in the class of K-?-bonvex function already existing in literature. Weak, strong and converse duality theorems are then established under higher-order K-?-convexity assumptions. Self duality is obtained by assuming the functions involved to be skew-symmetric. Several known results are also discussed as special cases.

A New Approach on Mixed-Type Nondifferentiable Higher-Order Symmetric Duality

In this paper, a new mixed-type higher-order symmetric duality in scalar-objective programming is formulated. In the literature we have results either Wolfe or Mond–Weir-type dual or separately, while in this we have combined those results over one model. The weak, strong and converse duality theorems are proved for these programs under $$\eta $$ -invexity/ $$\eta $$ -pseudoinvexity assumptions. Self-duality is also discussed. Our results generalize some existing dual formulations which were discussed by Agarwal et al. (Generalized second-order mixed symmetric duality in nondifferentiable mathematical programming. Abstr. Appl. Anal. 2011. https://doi.org/10.1155/2011/103597 ), Chen (Higher-order symmetric duality in nonlinear nondifferentiable programs), Gulati and Gupta (Wolfe type second order symmetric duality in nondifferentiable programming. J. Math. Anal. Appl. 310, 247–253, 2005, Higher order nondifferentiable symmetric duality with generalized F-convexity. J. Math. Anal. Appl. 329, 229–237, 2007), Gulati and Verma (Nondifferentiable higher order symmetric duality under invexity/generalized invexity. Filomat 28(8), 1661–1674, 2014), Hou and Yang (On second-order symmetric duality in nondifferentiable programming. J Math Anal Appl. 255, 488–491, 2001), Verma and Gulati (Higher order symmetric duality using generalized invexity. In: Proceeding of 3rd International Conference on Operations Research and Statistics (ORS). 2013. https://doi.org/10.5176/2251-1938_ORS13.16 , Wolfe type higher order symmetric duality under invexity. J Appl Math Inform. 32, 153–159, 2014).

Higher-order symmetric duality in multiobjective programming problems

In this paper, a pair of Mond-Weir type higher-order symmetric dual programs over arbitrary cones is formulated. The appropriate duality theorems, such as weak duality theorem, strong duality theorem and converse duality theorem, are established under higher-order (strongly) cone pseudoinvexity assumptions.

Higher-order symmetric duality with higher-order generalized invexity

Two different pairs of higher-order symmetric dual programs such as Wolfe type and Mond–Weir type are studied. The weak, strong and converse duality theorems are established for the higher-order symmetric dual programs under higher-order \(\rho -(\eta ,\theta )-\)invexity and \(\rho -(\eta ,\theta )-\)pseudo-invexity assumptions. Many examples and counterexamples are illustrated to justify our work. We also observe that several known results are obtained as special cases.

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.

A note on higher-order symmetric duality

In this paper, a pair of Wolfe type higher-order symmetric dual multiobjective programs is formulated. Strong and converse duality theorems are established under invexity assumptions. Duality relations for Mond–Weir type dual models have also been obtained under pseudoinvexity assumptions.

Higher-order symmetric duality under cone-invexity and other related concepts

In this paper we establish weak, strong and converse duality results for a pair of Wolfe type higher-order symmetric dual problems over cones under the assumption of higher-order cone-invexity. We also introduce the concepts of higher-order strictly and strongly cone-pseudoinvexity and use them to obtain weak, strong and converse duality results for the pair of Mond–Weir type higher-order symmetric dual problems.

Higher order symmetric duality in nondifferentiable multiobjective programming problems involving cones

In this paper, strong duality and converse duality theorems are established for a pair of Mond-Weir type multiobjective higher order symmetric dual programs over arbitrary cones. Our results ll some gaps in works of Agarwal et al. (2010) and Gupta et al. (2010).

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].

Higher-order symmetric duality for a class of multiobjective fractional programming problems

In this paper, a pair of nondifferentiable multiobjective fractional programming problems is formulated. For a differentiable function, we introduce the definition of higher-order (F, α, ρ, d)-convexity, which extends some kinds of generalized convexity, such as second order F-convexity and higher-order F -convexity. Under the higher-order (F, α, ρ, d)-convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems.Mathematics Subject Classification (2010) 90C29; 90C30; 90C46.

Higher-order symmetric duality in multiobjective programming problems under higher-order invexity

Abstract In this paper we present a pair of Wolfe and Mond–Weir type higher-order symmetric dual programs for multiobjective symmetric programming problems. Different types of higher-order duality results (weak, strong and converse duality) are established for the above higher-order symmetric dual programs under higher-order invexity and higher-order pseudo-invexity assumptions. Also we discuss many examples and counterexamples to justify our work.

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].

Multiobjective higher-order symmetric duality involving generalized cone-invex functions

In this paper, a pair of Mond–Weir type multiobjective higher-order symmetric dual programs over arbitrary cones is formulated and usual duality results are established under higher-order K -preinvexity/ K -pseudoinvexity assumptions. Symmetric minimax mixed integer primal and dual problems are also discussed.

Higher order symmetric duality in nondifferentiable multi-objective programming problems involving generalized cone convex functions

A pair of Mond–Weir type multi-objective higher order symmetric dual programs over arbitrary cones is formulated. Weak, strong and converse duality theorems are established under higher order K – F -convexity assumptions. Our results generalize several known results in the literature.

Higher-order symmetric duality in vector optimization problem involving generalized cone-invex functions

In this paper, a pair of higher-order symmetric dual model in vector optimization problem is formulated. The higher-order cone-pseudoinvex and higher-order strongly cone-pseudoinvex functions are defined. The weak, strong and converse duality theorems are established using these defined functions.

Higher-order symmetric duality with cone constraints

A pair of higher-order Wolfe and Mond–Weir type symmetric dual models with cone constraints are formulated and usual duality theorems are established under higher-order η -invexity/ η -pseudoinvexity assumptions. Symmetric minimax mixed integer primal and dual problems are also discussed. These duality relations also remove certain inconsistencies in some of the earlier results.