Fractional Multi-objective Optimization Problems Research Articles

Erratum to: Necessary optimality conditions for a fractional multiobjective optimization problem

Erratum to: RAIRO-Oper. Res. https://doi.org/10.1051/ro/2020049

Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem

<p style='text-indent:20px;'>In this work, we are concerned with a fractional multiobjective optimization problem <inline-formula><tex-math id="M1">\begin{document}$ (P) $\end{document}</tex-math></inline-formula> involving set-valued maps. Based on necessary optimality conditions given by Gadhi et al. [<xref ref-type="bibr" rid="b14">14</xref>], using support functions, we derive sufficient optimality conditions for <inline-formula><tex-math id="M2">\begin{document}$ \left( P\right) , $\end{document}</tex-math></inline-formula> and we establish various duality results by associating the given problem with its Mond-Weir dual problem <inline-formula><tex-math id="M3">\begin{document}$ \left( D\right) . $\end{document}</tex-math></inline-formula> The main tools we exploit are convexificators and generalized convexities. Examples that illustrates our findings are also given.</p>

Characterizations of Approximate Optimality Conditions for Fractional Semi-Infinite Optimization Problems With Uncertainty

This paper is devoted to the investigation of a class of multi-objective fractional semi-infinite optimization problems with uncertain data. Firstly, a robust counterpart for the uncertain multi-objective optimization problem is introduced in terms of robust optimization method. And following the idea due to Dinkelbach, we associate the robust counterpart with a multi-objective optimization problem. Then, by using a scalarization method, its corresponding scalarization optimization problem is established. The relationship between robust solutions of the multi-objective optimization problem and that of its corresponding scalarization optimization problem is described. Finally, by using a robust type sub-differential constraint qualification, the robust optimality condition for approximate quasi-efficient solutions of the multi-objective fractional optimization problem is established.

Necessary optimality conditions for a fractional multiobjective optimization problem

In this paper, we are concerned with a fractional multiobjective optimization problem (P). Using support functions together with a generalized Guignard constraint qualification, we give necessary optimality conditions in terms of convexificators and the Karush–Kuhn–Tucker multipliers. Several intermediate optimization problems have been introduced to help us in our investigation.

A Branch–Bound Cut Technique for Non-linear Fractional Multi-objective Optimization Problems

This article establishes a branch–bound technique to solve nonlinear convex–convex fractional multi-objective optimization problem in the non-convex feasible region. As far as the authors are concerned, this kind of problem is not solved by any other author in the literature. By transformation, multi-objective non-linear fractional problem is transformed into a multi-objective non-linear optimization problem. After giving preferences of weight to each objective, the original NLFMOOP is transformed into a nonlinear single-objective programming problem. Lagrange’s theorem of weak duality is used to find lower and upper bound for single objective nonlinear optimization problems in the feasible region. Some theoretical results for solving the multi-objective non-linear fractional problem have also been established. For showing the application of the proposed method, it has been applied to two numerical problems.

Optimality criteria for fuzzy-valued fractional multi-objective optimization problem

This paper presents optimality criteria for fuzzy-valued fractional multi-objective optimization problem. There are numerous optimality criteria which have been established for the deterministic fractional multi-objective optimization problems. Very few studies are available on the establishment of optimality criteria for fuzzy-valued multi-objective optimization problem. So, Karush–Kuhn–Tucker optimality criteria for fuzzy-valued fractional multi-objective problem are established by using Lagrange multipliers. First, the original problem is modified using the parametric approach of Dinkelbach into multi-objective non-fractional optimization problem, and then, the optimality conditions are established for the modified problem using the Hukuhara derivative. The established optimality criteria are verified by two numerical examples.

Generalized asymptotic functions in nonconvex multiobjective optimization problems

In this paper, we use generalized asymptotic functions and second-order asymptotic cones to develop a general existence result for the nonemptiness of the proper efficient solution set and a sufficient condition for the domination property in nonconvex multiobjective optimization problems. A new necessary condition for a point to be efficient or weakly efficient solution is given without any convexity assumption. We also provide a finer outer estimate for the asymptotic cone of the weakly efficient solution set in the quasiconvex case. Finally, we apply our results to the linear fractional multiobjective optimization problem.

Some characterizations of robust optimal solutions for uncertain fractional optimization and applications

In this paper, following the framework of robust optimization, we consider robust optimal solutions for a fractional optimization problem in the face of data uncertainty both in the objective and constraints. To this end, by using the properties of the subdifferential sum formulae, we first introduce some robust basic subdifferential constraint qualifications, and then obtain some completely characterizations of the robust optimal solutions of this uncertain fractional optimization problem. We show that our results encompass as special cases some optimization problems considered in the recent literature. Moreover, as applications, the proposed approach is applied to investigate weakly robust efficient solutions for multi-objective fractional optimization problems in the face of data uncertainty both in the objective and constraints.

Nondifferentiable fractional semi-infinite multiobjective optimization problems

Employing some advanced tools of variational analysis and generalized differentiation, we establish necessary conditions for local (weakly) efficient solutions of a nonsmooth fractional multiobjective optimization problem with an infinite number of inequality constraints. Sufficient conditions for such solutions to the considered problem are also obtained by means of proposing the use of (strictly) generalized convex functions. In addition, we state a dual problem to the primal one and explore duality relations.

A class of nonsmooth fractional multiobjective optimization problems

This paper focuses on the study of optimality conditions and duality in nonsmooth fractional multiobjective optimization problems. Applying some advanced tools of variational analysis and generalized differentiation, we establish necessary optimality conditions for (weakly) efficient solutions of a fractional multiobjective optimization problem involving inequality and equality constraints. Sufficient optimality conditions for such solutions to the considered problem are also obtained by means of (strictly) generalized convex-affine functions. In addition, we address a dual problem to the primal one and examine duality relations between them.

Linear fractional multi-objective optimization problems subject to fuzzy relational equations with the max-average composition

‎In this paper‎, ‎linear fractional multi-objective optimization problems subject to a system of fuzzy relational equations (FRE) using the max-average composition are considered‎. ‎First‎, ‎some theorems and results are presented to thoroughly identify and reduce the feasible set of the fuzzy relation equations‎. ‎Next‎, ‎the linear fractional multi-objective optimization problem is converted to a linear one using Nykowski and Zolkiewski's approach‎. ‎Then‎, ‎the efficient solutions are obtained by applying the improved e-constraint method‎. ‎‎Finally‎, ‎the proposed method is effectively tested by solving a consistent test problem‎.

Optimality criteria for sum of fractional multiobjective optimization problem with generalized invexity

The sum of a fractional program is a nonconvex optimization problem in the field of fractional programming and it is difficult to solve. The development of research is restricted to single objective sums of fractional problems only. The branch and bound methods/algorithms are developed in the literature for this problem as a single objective problem. The theoretical and algorithmic development for sums of fractional programming problems is restricted to single objective problems. In this paper, some new optimality conditions are proposed for the sum of a fractional multiobjective optimization problem with generalized invexity. The optimality conditions are obtained by using a modified objective approach and equivalency with the original problem is established.

Linear fractional multi-objective optimization problems subject to fuzzy relational equations with a continuous Archimedean triangular norm

In this paper, linear fractional multi-objective optimization problems subject to a system of fuzzy relational equations (FREs) using the max-Archimedean triangular norm composition are considered. First, some theorems and results are presented to simplify systems of fuzzy relational equations. Next, the feasible set of the optimization problem is reduced. Then, the linear fractional multi-objective optimization problem is converted to a linear one using Nykowski and Zolkiewski’s approach. Furthermore, the efficient solutions are obtained by applying the improved ε-constraint method.In addition, we design an algorithm to generate consistent systems of fuzzy relational equations, randomly, using the covering problem. Finally, the proposed method is effectively tested by solving a randomly generated consistent test problem.

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.

On ε-optimality conditions for multiobjective fractional optimization problems

Abstract A multiobjective fractional optimization problem (MFP), which consists of more than two fractional objective functions with convex numerator functions and convex denominator functions, finitely many convex constraint functions, and a geometric constraint set, is considered. Using parametric approach, we transform the problem (MFP) into the non-fractional multiobjective convex optimization problem (NMCP) v with parametric v ∈ ℝ p , and then give the equivalent relation between (weakly) ε-efficient solution of (MFP) and (weakly) "Equation missing"-efficient solution of "Equation missing". Using the equivalent relations, we obtain ε- optimality conditions for (weakly) ε- efficient solution for (MFP). Furthermore, we present examples illustrating the main results of this study. 2000 Mathematics Subject Classification: 90C30, 90C46.

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.

Comparison between different duals in multiobjective fractional programming

The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem expressed in terms of the conjugates of the functions involved. The weak, strong and converse duality statements for the intermediate problems allow us to give dual characterizations for the efficient solutions of the initial fractional problem. The aim of this paper is to compare the intermediate dual problem with other similar dual problems known from the literature. We completely establish the inclusion relations between the image sets of the duals as well as between the sets of maximal elements of the image sets.

On optimality and duality for nonsmooth multiobjective fractional optimization problems

In this paper, we consider a nonsmooth multiobjective fractional program. For sufficient optimality conditions, we define the nearly invex functions for locally Lipschitz vector-valued functions. We obtain generalized sufficient optimality theorems and prove weak and strong duality theorems for the multiobjective fractional optimization problem involving nearly invex functions.

Efficiency Criteria and Duality Models for Multiobjective Fractional Programming Problems Containing Locall'y Subdifferentiable and ρ-Convex Functions

Both parametric and semiparametric stationary-point- and saddle-point-type efficiency conditions are established for a lass of multiobjective fractional programming problems with subdifferentiable and ρ-convex functions. Subsequently, these efficiency criteria are utilized as a basis for constructing a Lagrangian-type and several Wolfe-type parametric and semiparametric duality models and proving appropriate duality theorems. To illustrate the relevance and applicability of these results, several classes of multiobjective fractional optimization problems which can be viewed as special cases of the main problem are briefly discussed. These include problems with arbitrary norms, square roots of positive semidefinite quadratic forms, support functions, infinitely many constraints, continuous max functions, and discrete max functions.