Nondifferentiable Multiobjective Programming Problem Research Articles

Optimality conditions and duality for nondifferentiable multiobjective programming problems involving d- r-type I functions

In this paper, new classes of nondifferentiable functions constituting multiobjective programming problems are introduced. Namely, the classes of d - r -type I objective and constraint functions and, moreover, the various classes of generalized d - r -type I objective and constraint functions are defined for directionally differentiable multiobjective programming problems. Sufficient optimality conditions and various Mond–Weir duality results are proved for nondifferentiable multiobjective programming problems involving functions of such type. Finally, it is showed that the introduced d - r -type I notion with r ≠ 0 is not a sufficient condition for Wolfe weak duality to hold. These results are illustrated in the paper by suitable examples.

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 R n . 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.

Optimality and duality in nondifferentiable multiobjective mathematical programming involving higher order (F,α,ρ,d)-type I functions

In this paper, a nondifferentiable multiobjective programming problem is considered where every component of objective and constraint functions contain a term involving the support function of a compact convex set. A new class of higher order (F,α,ρ,d)-type I function is introduced. Necessary optimality conditions and the duality theorems for Wolfe and unified higher order dual problems are established. Several known results can be deduced as special cases.

Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs

In this paper, a class of nondifferentiable multiobjective fractional programs is studied, in which every component of the objective function contains a term involving the support function of a compact convex set. Kuhn-Tucker necessary and sufficient optimality conditions, duality and saddle point results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problems are given. The results presented in this paper improve and extend some the corresponding results in the literature.

Second-Order Duality for Nondifferentiable Multiobjective Programming Problems

This paper is concerned with second-order duality for a class of nondifferentiable multiobjective programming problems. Usual duality theorems are proved for Mangasarian type and general Mond–Weir type vector duals under generalized bonvexity assumptions.

Higher-Order Duality in Nondifferentiable Multiobjective Programming

A new generalized class of higher-order (F, α, ρ, d)–type I functions is introduced, and a general Mond–Weir type higher-order dual is formulated for a nondifferentiable multiobjective programming problem. Based on the concepts introduced, various higher-order duality results are established. At the end, some special cases are also discussed.

Mixed symmetric duality in non-differentiable multiobjective mathematical programming

Two mixed symmetric dual models for a class of non-differentiable multiobjective nonlinear programming problems with multiple arguments are introduced in this paper. These two mixed symmetric dual models unify the four existing multiobjective symmetric dual models in the literature. Weak and strong duality theorems are established for these models under some mild assumptions of generalized convexity. Several special cases are also obtained.

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.

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,?,?,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.

Second Order Mixed Symmetric Duality in Non-Differentiable Multi-Objective Mathematical Programming

A pair of Mond-Weir type second order mixed symmetric duals is presented for a class of non-differentiable multi-objective non-linear programming problems with multiple arguments. We establish duality theorems for the new pair of dual models under second order generalized convexity assumptions. This mixed second order dual formulation unifies the two existing second order symmetric dual formulations in the literature. Many recent works on symmetric duality are obtained as special cases of the results established in the present paper.

Optimality and Duality for a Class of Nondifferentiable Multiobjective Fractional Programming Problems

In this paper, we consider a class of nondifferentiable multiobjective fractional programs in which each component of the objective function contains a term involving the support function of a compact convex set. We establish necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems.

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.

Some nondifferentiable multiobjective programming problems

In this paper it is shown that a relaxation defining the class of generalized d-V-type-I functions leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar non-differentiable case, and avoids the major difficulty of verifying that the inequality holds for the same kernel function. The results obtained in this paper generalize and extend the previously known results in this area.

Second order symmetric duality in non-differentiable multiobjective programming with F-convexity

This paper is concerned with a pair of Mond–Weir type second order symmetric dual non-differentiable multiobjective programming problems. We establish the weak and strong duality theorems for the new pair of dual models under second order F-convexity assumptions. Several results including many recent works are obtained as special cases.

Nondifferentiable multiobjective programming under generalized d-univexity

In this paper, we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints. We introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions in Aghezzaf and Hachimi [Numer. Funct. Anal. Optim. 22 (2001) 775], d-invex functions in Antczak [Europ. J. Oper. Res. 137 (2002) 28] and univex functions in Bector et al. [Univex functions and univex nonlinear programming, Proc. Admin. Sci. Assoc. Canada, 1992, p. 115]. By utilizing the new concepts, we derive a Karush–Kuhn–Tucker sufficient optimality condition and establish Mond–Weir type and general Mond–Weir type duality results for the nondifferentiable multiobjective programming problem.

Optimality and Duality in Nondifferentiable and Multiobjective Programming under Generalized d-Invexity

In this paper, we are concerned with the nondifferentiable multiobjective programming problem with inequality constraints. We introduce four new classes of generalized d-type-I functions. By utilizing the new concepts, Antczak type Karush-Kuhn-Tucker sufficient optimality conditions, Mond-Weir type and general Mond-Weir type duality results are obtained for non-differentiable and multiobjective programming.

ε-optimality for multiobjective programming on a Banach space

We establish a Lagrange multiplier rule, in terms of Clarke generalized gradient, that characterizes ε-efficiency of a nondifferentiable multiobjective programming problem on a real Banach space. This rule is then utilized to establish relationship between ε-efficient solution of multiobjective program and generalized saddle point of an appropriate vector-valued Lagrangian under invexity hypothesis.

Higher-order symmetric duality in nondifferentiable multiobjective programming problems

In this paper, a pair of nondifferentiable multiobjective programming problems is first formulated, where each of the objective functions contains a support function of a compact convex set in R n . For a differentiable function h :R n×R n→R , we introduce the definitions of the higher-order F-convexity ( F-pseudo-convexity, F-quasi-convexity) of function f :R n→R with respect to h. When F and h are taken certain appropriate transformations, all known other generalized invexity, such as η-invexity, type I invexity and higher-order type I invexity, can be put into the category of the higher-order F-invex functions. Under these the higher-order F-convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems related to a properly efficient solution.

Symmetric duality for a class of multiobjective fractional programming problems

This paper is concerned with symmetric duality for a class of nondifferentiable multiobjective fractional programming problems. Two weak duality theorems and two strong duality theorems are proved. Discussion on some special cases shows that results in this paper extend previous work in this area.

Duality for a Class of Nondifferentiable Multiobjective Programming Problems

In this paper, both the Wolfe type and Mond–Weir type dual problems for a class of nondifferentiable multiobjective programs in which every component of the objective function contains a term involving the support function of a compact convex set are formulated. Weak duality theorems are established under suitable conditions. A special case which appears repeatedly in the literature is that the support function is the square root of a positive semidefinite quadratic form. This and other special cases can be readily generated from our results.