Nondifferentiable Multiobjective Fractional Programming Problem Research Articles

Optimality, duality and saddle point analysis for interval-valued nondifferentiable multiobjective fractional programming problems

This paper is developed to discuss the theoretical aspects of multiobjective fractional problems under interval uncertainty. For this, we have formulated an interval multiobjective fractional model (IVMFP) in which the numerator and denominator of objective functions are assumed to be interval valued functions. Partial ordering is invoked to define LU-Pareto optimal solutions to the problem. Thereafter, necessary optimality conditions of the Fritz John type and Kuhn Tucker type are derived in the classical way. LU-convexity and LU-concavity are used to establish the sufficient optimality criteria and with the same assumptions, the parametric duality theory is discussed. Also, interval valued vector Lagrangian is proposed and the corresponding results are established to locate the saddle point.

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.

Nondifferentiable higher-order duality theorems for new type of dual model under generalized functions

The motivation behind this article is to study a class of nondifferentiable multiobjective fractional programming problem in which each component of objective functions contains a term including the support function of a compact convex set. For a differentiable function, we consider a class of higher order pseudo quasi/ strictly pseudo quasi/weak strictly pseudo quasi- (V, ρ, d)-type-I convex functions. Under these the higher-order pseudo quasi/ strictly pseudo quasi/weak strictly pseudo quasi- (V, ρ, d)-type-I convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems related to efficient solution.

Second-order nondifferentiable multiobjective mixed type fractional programming programming problems

The motivation behind this paper is to study a class of nondifferentiable multiobjective fractional programming problem in which each component of objective functions contains a term including the support function of a compact convex set. For a differentiable function, we consider the class of second-order (F,alpha,rho,d)-type-I convex functions. Further, we formulate unified (mixed type) dual models and derive duality relations under aforesaid assumptions.

Parametric nondifferentiable multiobjective fractional programming under (b;;; )-univexity

In this paper, we are concerned with optimality conditions and duality results for nondifferentiable multiobjective fractional programming problems. Parametric necessary optimality conditions are established for such vector optimization problems in which each component of the involved functions is locally Lipschitz. Further, under the introduced concept of nondifferentiable $(b,\Psi ,\Phi ,\rho )$-univexity, the parametric sufficient optimality conditions are established for a new class of nonconvex multiobjective fractional programming problems. Furthermore, for the considered multiobjective fractional programming problem, its parametric vector dual problem in the sense of Schaible is defined. Then several duality theorems are also established under $(b,\Psi ,\Phi ,\rho )$% -univexity hypotheses.

On duality for a second-order multiobjective fractional programming problem involving type-I functions

Abstract The purpose of this paper is to study a nondifferentiable multiobjective fractional programming problem (MFP) in which each component of objective functions contains the support function of a compact convex set. For a differentiable function, we introduce the class of second-order ( C , α , ρ , d ) - V {(C,\alpha,\rho,d)-V} -type-I convex functions. Further, Mond–Weir- and Wolfe-type duals are formulated for this problem and appropriate duality results are proved under the aforesaid assumptions.

Duality results for a second-order multiobjective fractional programming problem with generalised convexity

The purpose of this paper is to study a class of non-differentiable multiobjective fractional programming problem in which every component of objective functions contains a term involving the support function of a compact convex set. For a differentiable function, we use the definition of second-order (C, α, ρ, d) - V-type-I convex function. Further, a Mond-Weir type dual has been formulated for this problem and appropriate duality results have been proved under second-order (C, α, ρ, d) - V-type-I convexity assumptions.

Saddle Point Criteria in Multiobjective Fractional Programming Involving Exponential Invexity

Consider a system of nondifferentiable multiobjective nonlinear fractional programming problem involving exponential $$(p,\,r)$$ -invex functions with respect to $$\eta .$$ A new concept of saddle point for a mixed Lagrange function is introduced. We establish the equivalence between the saddle point and an efficient solution of the minimization multiobjective fractional problem involving exponential $$(p,\,r)$$ -invexity assumptions.

Optimality and duality results for a nondifferentiable multiobjective fractional programming problem

The purpose of this paper is to study a class of nondifferentiable multiobjective fractional programming problems in which every component of objective functions contains a term involving the support function of a compact convex set. For a differentiable function, we introduce the definition of higher-order $(C,\alpha,\gamma,\rho,d)$ -convex function. A nontrivial example is also constructed which is in this class but not $(F,\alpha,\gamma,\rho,d)$ -convex. Based on the $(C,\alpha,\gamma,\rho,d)$ -convexity, sufficient optimality conditions for an efficient solution of the nondifferentiable multiobjective fractional programming problem are established. Further, a higher-order Mond-Weir type dual is formulated for this problem and appropriate duality results are proved under higher-order $(C,\alpha,\gamma,\rho,d)$ -assumptions.

SECOND ORDER NONDIFFERENTIABLE MULTIOBJECTIVE FRACTIONAL PROGRAMMING UNDER GENERALIZED UNIVEX FUNCTIONS

In this paper, a new class of second order $(d,\rho,\eta,\theta)$-type 1 univex function is introduced. The Wolfe type second order dual problem (SFD) of the nondifferentiable multiobjective fractional programming problem (MFP) is considered, where the objective and constraint functions involved are directionally differentiable. Also the duality results under second order$(d,\rho,\eta,\theta)$-type 1 univex functions are established.

Second Order Duality in Multiobjective Fractional Programming with Square Root Term under Generalized Univex Function.

Three approaches of second order mixed type duality are introduced for a nondifferentiable multiobjective fractional programming problem in which the numerator and denominator of objective function contain square root of positive semidefinite quadratic form. Also, the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterization technique is used to establish duality results under generalized second order ρ-univexity assumption.

Mixed type duality in non-differentiable multi-objective fractional programming under generalised (Φ, ρ)-univex function

In this paper, a new class of generalised (Φ, ρ)-univex function is introduced. Three approaches for both primal and mixed type dual of a non-differentiable multi-objective fractional programming problems are presented. Also, the necessary and sufficient conditions of efficient solution for fractional programming are established. Weak and strong duality theorems are established under generalised (Φ, ρ)-univexity assumption.

Mixed type duality for nondifferentiable multiobjective fractional programming under generalized (d, ρ, η, θ)-type 1 univex function

In this paper, we introduced a new generalized class of (d,ρ,η,θ)-type 1 univex function with an example. We introduced the mixed duality problem (MFXD) of the nondifferentiable multiobjective fractional programming problem (MFP) where the objective and constraint functions involved are directionally differentiable. Also we established the duality results under generalized class of (d,ρ,η,θ)-type 1 univex function.

On second- order symmetric duality for a class of semigroups with zero deficiency

This article is concerned with a pair of second-order symmetric duals in the context of non-differentiable multiobjective fractional programming problems. We establish the weak and strong duality theorems for the new pair of dual models. Discussion on some special cases shows that results in this paper extend previous work in this area.

On second- order symmetric duality for a class of multiobjective fractional programming problem

This article is concerned with a pair of second-order symmetric duals in the context of non-differentiable multiobjective fractional programming problems. We establish the weak and strong duality theorems for the new pair of dual models. Discussion on some special cases shows that results in this paper extend previous work in this area.

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.

Sufficiency and duality in nondifferentiable multiobjective fractional programming with higher-order (V, α, ρ, θ)-invexity

In the present paper, the concept of higher-order (V , α , ρ , θ) invexity is used to study higher-order duality for a nondifferentiable multiobjective fractional programming problem (MFP). We first obtain a result giving higher-order (V, α, ρ, θ)-invexity of the ratio of two functions. This result is then used to drive sufficient optimality conditions for an efficient solution of (MFP). Moreover, duality theorems are established for Mond-Weir type higher-order dual of (MFP).

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.

ON OPTIMALITY AND DUALITY FOR GENERALIZED NONDIFFERENTIABLE FRACTIONAL OPTIMIZATION PROBLEMS

A generalized nondifferentiable fractional optimization problem (GFP), which consists of a maximum objective function defined by finite fractional functions with differentiable functions and support functions, and a constraint set defined by differentiable functions, is considered. Recently, Kim et al. [Journal of Optimization Theory and Applications 129 (2006), no. 1, 131-146] proved optimality theorems and duality theorems for a nondifferentiable multiobjective fractional programming problem (MFP), which consists of a vector-valued function whose components are fractional functions with differentiable functions and support functions, and a constraint set defined by differentiable functions. In fact if <TEX>$\overline{x}$</TEX> is a solution of (GFP), then <TEX>$\overline{x}$</TEX> is a weakly efficient solution of (MFP), but the converse may not be true. So, it seems to be not trivial that we apply the approach of Kim et al. to (GFP). However, modifying their approach, we obtain optimality conditions and duality results for (GFP).

DUALITY IN NONDIFFERENTIABLE MULTIOBJECTIVE FRACTIONAL PROGRAMS INVOLVING CONES

In this paper, we introduce nondifferentiable multiobjective fractional programming problems with cone constraints over arbitrary closed convex cones, 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 and strong duality theorems for a weakly efficient solution under suitable (V, ρ)-invexity assumptions. As special cases of our duality relations, we give some known duality results.