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.     

Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity

In this paper, we consider nondifferentiable multiobjective fractional programming problems. A concept of generalized convexity, which is called (C,α,ρ,d)-convexity, is first discussed. Based on this generalized convexity, we obtain efficiency conditions for multiobjective fractional programming (MFP). Furthermore, we establish duality results for three types of dual problems of (MFP) and present the corresponding duality theorems.     

Nondifferentiable Minimax Fractional Programming Problems with (C, α, ρ, d)-Convexity

In this paper, we present necessary optimality conditions for nondifferentiable minimax fractional programming problems. A new concept of generalized convexity, called (C, α, ρ, d)-convexity, is introduced. We establish also sufficient optimality conditions for nondifferentiable minimax fractional programming problems from the viewpoint of the new generalized convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for two types of dual programs. This research was partially supported by NSF and Air Force grants     

Sufficiency and duality for multiobjective control problems under generalized (<Emphasis Type="Italic">B</Emphasis>, <Emphasis Type="Italic">ρ</Emphasis>)-type I functions

In this paper, we introduce the classes of (B, ρ)-type I and generalized (B, ρ)-type I, and derive various sufficient optimality conditions and mixed type duality results for multiobjective control problems under (B, ρ)-type I and generalized (B, ρ)-type I assumptions.     

Optimality and duality for a nonsmooth multiobjective optimization involving generalized type I functions

A nonsmooth multiobjective optimization problem involving generalized (F, α, ρ, d)-type I function is considered. Karush–Kuhn–Tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be an efficient or properly efficient solution. Duality results are obtained for mixed type dual under the aforesaid assumptions.     

Sufficiency and duality in nondifferentiable multiobjective programming involving generalized type I functions

Recently, Hachimi and Aghezzaf defined generalized (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. In this paper, the generalized (F,α,ρ,d)-type I functions are extended to nondifferentiable functions. By utilizing the new concepts, we obtain several sufficient optimality conditions and prove mixed type and Mond-Weir type duality results for the nondifferentiable multiobjective programming problem.     

Optimality Conditions and Duality in Nondifferentiable Minimax Fractional Programming with Generalized Convexity

We establish sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems involving (F, α, ρ, d)-convexity. Subsequently, we apply the optimality conditions to formulate two types of dual problems and prove appropriate duality theorems. The authors thank the referee for valuable suggestions improving the presentation of the paper.     

Optimality Conditions and Duality for?Nondifferentiable Multiobjective Fractional Programming Problems with?(C,α,ρ,d)-convexity

The purpose of this paper is to consider a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function contains a term involving the support function of a compact convex set. Based on the (C,α,ρ,d)-convexity, sufficient optimality conditions and duality results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problem are established. The results extend and improve the corresponding results in the literature.     

Sufficiency in Nonsmooth Multiobjective Programming Involving Generalized (Fρ)-convexity

In this paper, we consider a generalization of convexity for nonsmooth multiobjective programming problems. We obtain sufficient optimality conditions under generalized (Fρ)-convexity.This work was supported by Project 821134 and by the Center of Excellence for Mathematics, University of Isfahan, Isfahan, Iran.Communicated by F. Giannessi     

Generalized V-univexity type-I for multiobjective programming with n-set functions

In the last time important results in multiobjective programming involving type-I functions were obtained (Yuan et al. in: Konnov et al. (eds) Lecture notes in economics and mathematical systems, 2007; Mishra et al. An Univ Bucureşti Ser Mat, LII(2): 207–224, 2003). Following one of these ways, we study optimality conditions and generalized Mond-Weir duality for multiobjective programming involving n-set functions which satisfy appropriate generalized univexity V-type-I conditions. We introduce classes of functions called (ρ, ρ′)-V-univex type-I, (ρ, ρ′)-quasi V-univex type-I, (ρ, ρ′)-pseudo V-univex type-I, (ρ, ρ′)-quasi pseudo V-univex type-I, and (ρ, ρ′)-pseudo quasi V-univex type-I. Finally, a general frame for constructing functions of these classes is considered. This research was supported by Grant PN II code ID No. 112/01.10.2007, CEEX code 1/2006 No. 1531/2006, and CNCSIS A No. 105 GR/2006.     

Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions

In this paper, new classes of generalized (F,α,ρ,d)-type I functions are introduced for differentiable multiobjective programming. Based upon these generalized functions, first, we obtain several sufficient optimality conditions for feasible solution to be an efficient or weak efficient solution. Second, we prove weak and strong duality theorems for mixed type duality.     

Optimality conditions and duality for a class of nondifferentiable multi-objective fractional programming problems

A class of multi-objective fractional programming problems (MFP) are considered where the involved functions are locally Lipschitz. In order to deduce our main results, we give the definition of the generalized (F,θ,ρ,d)-convex class about the Clarke’s generalized gradient. Under the above generalized convexity assumption, necessary and sufficient conditions for optimality are given. Finally, a dual problem corresponding to (MFP) is formulated, appropriate dual theorems are proved.      

Higher-Order Duality in Nondifferentiable Minimax Programming with Generalized Type I Functions

A unified higher-order dual for a nondifferentiable minimax programming problem is formulated. Weak, strong and strict converse duality theorems are discussed involving generalized higher-order (F,α,ρ,d)-Type I functions. The research of second author was supported by the Department of Atomic Energy, Government of India, under the NBHM Post Doctoral Fellowship Program 40/9/2005-R&D II/2398.     

Nonsmooth ρ ? (η, θ)-invexity in multiobjective programming problems

In this paper we extend Reiland’s results for a nonlinear (single objective) optimization problem involving nonsmooth Lipschitz functions to a nonlinear multiobjective optimization problem (MP) for ρ − (η, θ)-invex functions. The generalized form of the Kuhn–Tucker optimality theorem and the duality results are established for (MP).     

Higher order duality in multiobjective fractional programming with support functions

In this paper a new class of higher order (F,ρ,σ)-type I functions for a multiobjective programming problem is introduced, which subsumes several known studied classes. Higher order Mond-Weir and Schaible type dual programs are formulated for a nondifferentiable multiobjective fractional programming problem where the objective functions and the constraints contain support functions of compact convex sets in Rn. Weak and strong duality results are studied in both the cases assuming the involved functions to be higher order (F,ρ,σ)-type I. A number of previously studied problems appear as special cases.     

On sufficiency and duality in multiobjective programming problem under generalized <Emphasis Type="Italic">α</Emphasis>-type I univexity

In this paper, we are concerned with the multiobjective programming problem with inequality constraints. We introduce new classes of generalized α-univex type I vector valued functions. A number of Kuhn–Tucker type sufficient optimality conditions are obtained for a feasible solution to be an efficient solution. The Mond–Weir type duality results are also presented.     

Second order duality for nondifferentiable multiobjective programming problem involving (F, α, ρ, d)-V-type I functions

In this paper, new classes of second order (F, α, ρ, d)-V-type I functions for a nondifferentiable multiobjective programming problem are introduced. Furthermore, second order Mangasarian type and general Mond-Weir type duals problems are formulated for a nondifferentiable multiobjective programming problem. Weak strong and strict converse duality theorems are studied in both cases assuming the involved functions to be second order (F, α, ρ, d)-V-type I.     

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.     

Second order mixed type duality in multiobjective programming problems

Second order mixed type dual is introduced for multiobjective programming problems. Results about weak duality, strong duality, and strict converse duality are established under generalized second order (F,ρ)-convexity assumptions. These results generalize the duality results recently given by Aghezzaf and Hachimi involving generalized first order (F,ρ)-convexity conditions.     

Optimality for (<Emphasis Type="Italic">h,φ</Emphasis>)-multiobjective programming involving generalized type-I functions

Based upon Ben-Tal’s generalized algebraic operations, new classes of functions, namely (h,φ)-type-I, quasi (h,φ)-type-I, and pseudo (h,φ)-type-I, are defined for a multi-objective programming problem. Sufficient optimality conditions are obtained for a feasible solution to be a Pareto efficient solution for this problem. Some duality results are established by utilizing the above defined classes of functions, considering the concept of a Pareto efficient solution. This research is supported by National Science Foundation of China under Grant No. 69972036.