Multiobjective Fractional Problems Research Articles

On robust weakly $ \varepsilon $-efficient solutions for multi-objective fractional programming problems under data uncertainty

<abstract><p>In this study, we use the robust optimization techniques to consider a class of multi-objective fractional programming problems in the presence of uncertain data in both of the objective function and the constraint functions. The components of the objective function vector are reported as ratios involving a convex non-negative function and a concave positive function. In addition, on applying a parametric approach, we establish $ \varepsilon $-optimality conditions for robust weakly $ \varepsilon $-efficient solution. Furthermore, we present some theorems to obtain a robust $ \varepsilon $-saddle point for uncertain multi-objective fractional problem.</p></abstract>

Optimality and duality for nonsmooth multiobjective fractional problems using convexificators

Optimality and duality for nonsmooth multiobjective fractional problems using convexificators

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.

On the Mangasarian–Fromovitz constraint qualification and Karush–Kuhn–Tucker conditions in nonsmooth semi-infinite multiobjective programming

We discuss constraint qualifications in Karush–Kuhn–Tucker multiplier rules in nonsmooth semi-infinite multiobjective programming. A version of the Manganarian–Fromovitz constraint qualification is proposed, in terms of the Michel–Penot directional derivative and the Studniarski derivative of order p which is just the order of the directional Holder metric subregularity which is included also in this proposed qualification version. Using this qualification together with the Pshenichnyi–Levitin–Valadire property, we establish Karush–Kuhn–Tucker optimality conditions for Borwein-proper and firm solutions. We also compare in detail our qualification version with other usually-employed constraint qualifications. Applications to semi-infinite multiobjective fractional problems and minimax problems are provided.

Optimality conditions for multiobjective fractional programming, via convexificators

In this paper, the idea of convexificators is used to derive the Karush-Kuhn-Tucker necessary optimality conditions for local weak efficient solutions of multiobjective fractional problems involving inequality and equality constraints. In this regard, several well known constraint qualifications are generalized and relationships between them are investigated. Moreover, some examples are provided to clarify our results.

Higher-Order Duality Relations for Multiobjective Fractional Problems Involving Support Functions

In this article, we have introduced a new class of higher-order $$(K\times Q)$$ -F-type I functions. Further, we have formulated two higher-order dual models, Wolfe and Schaible type, for a multiobjective fractional programming problem over arbitrary cones and proved appropriate duality relations under the higher-order $$(K\times Q)$$ -F-type I assumption. Nontrivial examples are also discussed to validate the weak duality results obtained in the paper.

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.

A Taxonomy and Review of the Multi-Objective Fractional Programming (MOFP) Problems

Optimizing the ratios within the constraints is called fractional programming or ratio optimization problem . If one can optimize several ratios objectives simultaneously, then it is called multi-objective fractional problem (MOFP). This paper presents a review on multi-objective fractional programming (MOFP) problems. In contrast, this review is excluded various technical parts of fractional programming but it gives a road map of studies available in the literature. The MOFP is classified into two classes, first one is General MOFP and other one is multi-objective linear fractional programming (MOLFP). Then, these two classes are subclassified based on the basis of proposed algorithm and optimality criteria. The basic concepts of MOFP are described in brief and for the further research, references are given in the end. The main contribution of authors is review and arrange the available literature in a systematic way.

Fuzzy optimality conditions for fractional multi-objective problems

In this article, we are concerned with fractional multi-objective optimization problems. Since those problems are in general nonconvex problems even if the problem data are convex, using techniques from variational analysis especially the approximate extremal principle [B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren Series: Fundamental Principles of Mathematical Sciences, Vol. 330, Springer, Berlin, 2006; B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Grundlehren Series: Fundamental Principles of Mathematical Sciences, Vol. 331, Springer, Berlin, 2006], we develop fuzzy optimality conditions.

Fuzzy Optimality Conditions for Fractional Multiobjective Bilevel Problems Under Fractional Constraints

In this article, we are concerned with fractional multiobjective optimization problems. In order to derive optimality conditions, we consider a new single level problem [12], which is locally equivalent to the bilevel fractional multiobjective problem (P) at the optimal solution. Our approach consists of using another approach initiated by Mordukhovich [7, 8], which does not involve any convex approximations and convex separation arguments, called the extremal principle [5, 6, 9], for the study of necessary optimality conditions in fractional vector optimization.

Necessary and sufficient optimality conditions for fractional multi-objective problems

Using a special scalarization, we give necessary optimality conditions for fractional multiobjective optimization problems. Under a generalized invexity, sufficient optimality conditions are also given. All over the article, the data are assumed to be continuous but not necessarily Lipschitz.

OPTIMALITY AND DUALITY FOR MULTIOBJECTIVE FRACTIONAL PROBLEMS WITH r-INVEXITY

We establish necessary and sufficient conditions for efficiency of multiobjective fractional programming problems involving r-invex functions. Using the optimality conditions, we investigate the parametric type dual, Wolfe type dual and Mond-Weir type dual for multiobjective fractional programming problems concerning r-invexity. Some duality theorems are also proved for such problem in the framework of r-invexity.

NONSMOOTH MULTIOBJECTIVE FRACTIONAL PROGRAMMING WITH GENERALIZED INVEXITY

In this paper, we consider nonsmooth multiobjective fractional programming problems involving locally Lipschitz functions. We introduce the property of generalized invexity for fractional function. We present necessary optimality conditions, sufficient optimality conditions and duality relations for nonsmooth multiobjective fractional programming problems, which is for a weakly efficient solution under suitable generalized invexity assumptions.

A modified objective function method for solving nonlinear multiobjective fractional programming problems

A new method is used for solving nonlinear multiobjective fractional programming problems having V-invex objective and constraint functions with respect to the same function η. In this approach, an equivalent vector programming problem is constructed by a modification of the objective fractional function in the original nonlinear multiobjective fractional problem. Furthermore, a modified Lagrange function is introduced for a constructed vector optimization problem. By the help of the modified Lagrange function, saddle point results are presented for the original nonlinear fractional programming problem with several ratios. Finally, a Mond–Weir type dual is associated, and weak, strong and converse duality results are established by using the introduced method with a modified function. To obtain these duality results between the original multiobjective fractional programming problem and its original Mond–Weir duals, a modified Mond–Weir vector dual problem with a modified objective function is constructed.

MULTIOBJECTIVE FRACTIONAL PROGRAMMING WITH A MODIFIED OBJECTIVE FUNCTION

We consider multiobjective fractional programming prob- lems with generalized invexity. An equivalent multiobjective pro- gramming problem is formulated by using a modiflcation of the objective function due to Antczak. We give relations between a multiobjective fractional programming problem and an equivalent multiobjective fractional problem which has a modifed objective function. And we present modifled vector saddle point theorems.

Symmetric duality for multiobjective fractional variational problems with generalized invexity

The concept of symmetric duality for multiobjective fractional problems has been extended to the class of multiobjective variational problems. Weak, strong and converse duality theorems are proved under generalized invexity assumptions. A close relationship between these problems and multiobjective fractional symmetric dual problems is also presented.

Constrained Vector Valued Ratio Games and Generalized Subdifferentiable Multiobjective Fractional Minmax Programming

Certain constrained two-person zero-sum ratio game with vector pay — off is considered and its relation with a pair of subdifferentiable multiobjective fractional programming problems is discussed. Relationship between an equilibrium point of a vector valued game with a saddle point of associated subdifferentiable multiobjective fractional problems is studied.