Quasi Efficient Solutions Research Articles

On $$\varepsilon $$-quasi efficient solutions for fractional infinite multiobjective optimization problems with locally Lipschitz data

On $$\varepsilon $$-quasi efficient solutions for fractional infinite multiobjective optimization problems with locally Lipschitz data

Further results on quasi efficient solutions in multiobjective optimization

Further results on quasi efficient solutions in multiobjective optimization

Quasi Efficient Solutions and Duality Results in a Multiobjective Optimization Problem with Mixed Constraints via Tangential Subdifferentials

We take up a nonsmooth multiobjective optimization problem with tangentially convex objective and constraint functions. In employing a suitable constraint qualification, we formulate both necessary and sufficient optimality conditions for (local) quasi efficient solutions in terms of tangential subdifferentials. Furthermore, under generalized convexity assumptions, we state strong, weak and converse duality relations of Wolfe and Mond–Weir types. We give a number of examples to illustrate the new concepts and main results of this paper.

Approximate solutions for robust multiobjective optimization programming in Asplund spaces

In this paper, we study a nonsmooth/nonconvex multiobjective optimization problem with uncertain constraints in arbitrary Asplund spaces. We first provide necessary optimality condition in a fuzzy form for approximate weakly robust efficient solutions and then establish necessary optimality theorem for approximate weakly robust quasi-efficient solutions of the problem in the sense of the limiting subdifferential by exploiting a fuzzy optimality condition in terms of the Fréchet subdifferential. Sufficient conditions for approximate (weakly) robust quasi-efficient solutions to such a problem are also driven under the new concept of generalized pseudo convex functions. Finally, we address an approximate Mond-Weir-type dual robust problem to the reference problem and explore weak, strong, and converse duality properties under assumptions of pseudo convexity.

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.

A unified concept of approximate and quasi efficient solutions and associated subdifferentials in multiobjective optimization

In this paper, we introduce some new notions of quasi efficiency and quasi proper efficiency for multiobjective optimization problems that reduce to the most important concepts of approximate and quasi efficient solutions given up to now. We establish main properties and provide characterizations for these solutions by linear and nonlinear scalarizations. With the help of quasi efficient solutions, a generalized subdifferential of a vector mapping is introduced, which generates a number of approximate subdifferentials frequently used in optimization in a unifying way. The generalized subdifferential is related to the classical subdifferential of real functions by the method of scalarization. An application of generalized subdifferential to express optimality conditions for quasi efficient solutions is also given.

Optimality conditions for $${\varvec{\epsilon }}$$ ϵ -quasi solutions of optimization problems via $${\varvec{\epsilon }}$$ ϵ -upper convexificators with applications

The aim of this paper is to provide necessary and sufficient conditions for a point to be an $$\epsilon $$ -quasi solution of a scalar optimization problem via $$\epsilon $$ -convexificators. Then, the main results are applied in order to obtain necessary conditions for approximate solutions of a vector optimization problem with constraints, which, at its turn, provides necessary conditions for $$\epsilon $$ -quasi efficient solutions of a vector equilibrium problem with constraints.

Vector critical points and generalized quasi-efficient solutions in nonsmooth multi-objective programming

In this work, several extended approximately invex vector-valued functions of higher order involving a generalized Jacobian are introduced, and some examples are presented to illustrate their existences. The notions of higher-order (weak) quasi-efficiency with respect to a function are proposed for a multi-objective programming. Under the introduced generalization of higher-order approximate invexities assumptions, we prove that the solutions of generalized vector variational-like inequalities in terms of the generalized Jacobian are the generalized quasi-efficient solutions of nonsmooth multi-objective programming problems. Moreover, the equivalent conditions are presented, namely, a vector critical point is a weakly quasi-efficient solution of higher order with respect to a function.

Approximate quasi efficiency of set-valued optimization problems via weak subdifferential

This paper deals with the approximate quasi efficient solutions of set-valued optimization problems. We establish the necessary and sufficient optimality conditions for \(\epsilon e\)-quasi efficiency via weak subdifferential under some approximate cone convexity assumptions. We also study duality results of Wolfe and Mond-Weir types for Geoffrion \(\epsilon e\)-quasi efficiency of finite dimensional set-valued optimization problems.

Scalarizations for Approximate Quasi Efficient Solutions in Multiobjective Optimization Problems

In this paper, by reviewing two standard scalarization techniques, a new necessary and sufficient condition for characterizing \((\varepsilon ,\bar{\varepsilon })\)-quasi (weakly) efficient solutions of multiobjective optimization problems is presented. The proposed procedure for the computation of \((\varepsilon ,\bar{\varepsilon })\)-quasi efficient solutions is given. Note that all of the provided results are established without any convexity assumptions on the problem under consideration. And our results extend several corresponding results in multiobjective optimization.

Optimality via generalized approximate convexity and quasiefficiency

In this article, we look beyond convexity and introduce the four new classes of functions, namely, approximate pseudoconvex functions of type I and type II and approximate quasiconvex functions of type I and type II. Suitable examples illustrating the non emptiness of the newly defined classes and distinguishing them from the existing classical notions of pseudoconvexity and quasiconvexity are provided. These newly defined concepts are then employed to establish sufficient optimality conditions for the quasi efficient solutions of a vector optimization problem.

Optimality conditions and duality on approximate solutions in vector optimization with arcwise connectivity

In this paper a new class of generalized vector-valued arcwise connected functions, termed sub-arcwise connected functions, is introduced. The properties of sub-arcwise connected functions are derived. The approximate quasi efficient solutions of vector optimization problems are studied, and the necessary and sufficient optimality conditions are obtained under the assumption of arcwise connectivity. An approximate Mond-Weir type dual problem is formulated and the duality theorems are established.

QUASIEFFICIENT SOLUTIONS OF MULTICRITERIA LOCATION PROBLEMS WITH RECTILINEAR NORM IN R^3

A multicriteria location problem with rectilinear norm in R 3 and its quasiefficient solutions are considered. First, we give characterizations of its quasiefficient solutions by using optimal solutions of a minisum location problem and by using the concept of the summary diagram. Next, we propose the Frame Generating Algorithm to find all quasiefficient solutions of the multicriteria location problem.

Approximate convexity in vector optimisation

Approximate convex functions are characterised in terms of Clarke generalised gradient. We apply this characterisation to derive optimality conditions for quasi efficient solutions of nonsmooth vector optimisation problems. Two new classes of generalised approximate convex functions are defined and mixed duality results are obtained.

On Efficient Solutions to Multiple Objective Mathematical Programs

This note develops properties of quasi-efficient solutions and explores interrelationships to the classical concept of efficiency. In particular, a point is a quasi-efficient solution to a multiple objective mathematical program if and only if it is an optimal solution to a scalar maximum problem for some set of nonnegative weights on the objectives. This result is then used to characterize the set of quasi-efficient solutions as the union of efficient solutions to a multiple objective problem over all nonempty subsets of the objectives.