Proper Efficient Solutions Research Articles

Characterization of E-Benson proper efficient solutions of vector optimization problems with variable ordering structures in linear spaces

Characterization of E-Benson proper efficient solutions of vector optimization problems with variable ordering structures in linear spaces

Optimality Conditions for Approximate Solutions of Set Optimization Problems with the Minkowski Difference

In this paper, we study the optimality conditions for set optimization problems with set criterion. Firstly, we establish a few important properties of the Minkowski difference for sets. Then, we introduce the generalized second-order lower radial epiderivative for a set-valued maps by Minkowski difference, and discuss some of its properties. Finally, by virtue of the generalized second-order lower radial epiderivatives and the generalized second-order radial epiderivatives, we establish the necessary optimality conditions and sufficient optimality conditions of approximate Benson proper efficient solutions and approximate weakly minimal solutions of unconstrained set optimization problems without convexity conditions, respectively. Some examples are provided to illustrate the main results obtained.

Higher-order optimality conditions of robust Benson proper efficient solutions in uncertain vector optimization problems

Higher-order optimality conditions of robust Benson proper efficient solutions in uncertain vector optimization problems

Modification of some scalarization approaches for multiobjective optimization

In this paper, we propose revisions of two existing scalarization approaches, namely the feasible-value constraint and the weighted constraint. These methods do not easily provide results on proper efficient solutions of a general multiobjective optimization problem. By proposing some novel modifications for these methods, we derive some interesting results concerning proper efficient solutions. These scalarization approaches need no convexity assumption of the objective functions. We also demonstrate the efficiency of the proposed method using numerical experiments. In particular, a rocket injector design problem involving four objective functions illustrates the performance of the proposed method.

Globally proper efficiency of set optimization problems based on the certainly set less order relation

In this paper, we investigate the globally proper efficiency of set optimization problems. Firstly, we use the so-called certainly set less order relation to define a new kind of set order relation. Based on the new set order relation, we introduce the notion of the globally proper efficient solution of the set optimization problem. Secondly, we establish Lagrange multiplier rule of the set optimization problem. Finally, we obtain Lagrangian duality theorems and saddle point theorems. We also give some examples to illustrate our results.

On Second-Order Semi-Differentiability of Index γ of Perturbation Maps in Parametric Vector Optimization Problems

In this paper, we prove that the Henig global proper efficient solution map and the Henig global proper efficient perturbation map of a parametric vector optimization problem are second-order semi-differentiable with index [Formula: see text] under some suitable qualification conditions. Several examples are given to illustrate the obtained results.

Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints

This paper discusses optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints. A new notion in terms of contingent cone and upper directional derivative is introduced, and a necessary condition for the Borwein proper efficient solution of the considered problem is derived. The concept of ε proper Abadie data qualification is also introduced, and a necessary condition which is called a strictly strong stationary condition for Borwein proper efficient solutions is obtained. In view of the strictly strong stationary condition, convexity of the objective functions, and quasi-convexity of constrained functions, sufficient conditions for the Borwein proper efficient solutions are presented. Some examples are given to illustrate the reasonability of the obtained results.

Scalarization and Optimality Conditions of E-Globally Proper Efficient Solution for Set-Valued Equilibrium Problems

In this paper, our purpose is to use the improvement set to investigate the scalarization and optimality conditions of E-globally proper efficient solution for the set-valued equilibrium problems with constraints. First, the notion of E-globally proper efficient solution for set-valued equilibrium problems with constraints is introduced in locally convex Hausdorff topological spaces. Second, the linear scalarization theorems of E-globally proper efficient solution are derived. Finally, under the assumption of nearly E-subconvexlikeness, the Kuhn–Tucker and Lagrange optimality conditions for set-valued equilibrium problems with constraints are obtained in the sense of E-globally proper efficiency. Meanwhile, we give some examples to illustrate our results. The results obtained in this paper improve and generalize some known results in the literature.

Characterizing quasiconvexity of the pointwise infimum of a family of arbitrary translations of quasiconvex functions, with applications to sums and quasiconvex optimization

It is well-known that the sum of two quasiconvex functions is not quasiconvex in general, and the same occurs with the minimum. Although apparently these two statements (for the sum or minimum) have nothing in common, they are related, as we show in this paper. To develop our study, the notion of quasiconvex family is introduced, and we establish various characterizations of such a concept: one of them being the quasiconvexity of the pointwise infimum of arbitrary translations of quasiconvex functions in the family; another is the convexity of the union of any two of their sublevel sets; a third one is the quasiconvexity of the sum of the quasiconvex functions, composed with arbitrary nondecreasing functions. As a by-product, any of the aforementioned characterizations, besides providing quasiconvexity of the sum, also implies the semistrict quasiconvexity of the sum if every function in the family has the same property. Three concrete applications in quasiconvex optimization are presented: First, we establish the convexity of the (Benson) proper efficient solution set to a quasiconvex vector optimization problem; second, we derive conditions that allow us to reduce a constrained optimization problem to one with a single inequality constraint, and finally, we show a class of quasiconvex minimization problems having zero duality gap.

The connectedness of the solutions set for set-valued vector equilibrium problems under improvement sets

In this paper, we provide the connectedness of the sets of weak efficient solutions, Henig efficient solutions and Benson proper efficient solutions for set-valued vector equilibrium problems under improvement sets.

On Second-Order Composed Proto-Differentiability of Proper Perturbation Maps in Parametric Vector Optimization Problems

The main aim of this paper is to study second-order sensitivity analysis in parametric vector optimization problems. We prove that the proper perturbation maps and the proper efficient solution maps of parametric vector optimization problems are second-order composed proto-differentiable under some appropriate qualification conditions. Some examples are provided to illustrate our results.

Painlevé–Kuratowski Convergence of Solutions for Perturbed Symmetric Set-Valued Quasi-Equilibrium Problem via Improvement Sets

This paper is devoted to study the Painlevé–Kuratowski convergence of solution sets for perturbed symmetric set-valued quasi-equilibrium problems (SSQEP)[Formula: see text] via improvement sets. By virtue of the oriented distance function, the sufficient conditions of Painlevé–Kuratowski convergence of efficient solution sets for (SSQEP)[Formula: see text] are obtained through a new nonlinear scalarization technical. Then, under [Formula: see text]-convergence of set-valued mappings, the Painlevé–Kuratowski convergence of weak efficient solution sets for (SSQEP)[Formula: see text] is discussed. What’s more, with suitable convergence assumptions, we also establish the sufficient conditions of lower Painlevé–Kuratowski convergence of Borwein proper efficient solution sets for (SSQEP)[Formula: see text] under improvement sets. Some interesting examples are formulated to illustrate the significance of the main results.

Approximate Proper Solutions in Vector Equilibrium Problems

The paper concerns with properties of approximate Benson and Henig proper efficient solutions of vector equilibrium problems. Relationships between both kinds of solutions and the approximate weak efficient solutions of the problem are stated. Roughly speaking, it is proved that they coincide provided that the ordering cone of the problem can be separated from another closed cone by an approximating sequence of dilating cones. As a result, the limit behavior of approximate Benson and Henig proper efficient solutions when the error tends to zero is studied. It is shown that efficient and proper efficient solutions of the problem can be obtained as the limit of approximate Benson and Henig proper efficient solutions whenever a suitable domination set is considered. Finally, optimality conditions for approximate Benson and Henig proper efficient solutions are derived by approximate solutions of scalar equilibrium problems. They are formulated via linear scalarization and the necessary conditions hold true in problems satisfying a kind of nearly subconvexlikeness condition. The main results of the paper generalize some recent ones because they involve weaker assumptions.

New second-order radial epiderivatives and applications to optimality conditions

In this paper, we introduce the second-order weakly composed radial epiderivative of set-valued maps, discuss its relationship to the second-order weakly composed contingent epiderivative, and obtain some of its properties. Then we establish the necessary optimality conditions and sufficient optimality conditions of Benson proper efficient solutions of constrained set-valued optimization problems by means of the second-order epiderivative. Some of our results improve and imply the corresponding ones in recent literature.

On Set Containment Characterizations for Sets Described by Set-Valued Maps with Applications

In this paper, dual characterizations of the containment of two sets involving convex set-valued maps are investigated. These results are expressed in terms of the epigraph of a conjugate function of infima associated with corresponding set-valued maps. As an application, we establish characterizations of weak and proper efficient solutions of set-valued optimization problems in the sense of vector criteria.

Limit Behavior of Approximate Proper Solutions in Vector Optimization

In the framework of a vector optimization problem, we provide conditions for approximate proper solutions to tend to exact weak/efficient/proper solutions when the error tends to zero. This limit behavior depends on an approximation set that is used to define the approximate proper efficient solutions. We also study the special case when the final space of the vector optimization problem is normed, and more particularly, when it is finite dimensional. In these specific frameworks, we provide several explicit constructions of dilating ordering cones and approximation sets that lead to the desired limit behavior. In proving our results, new relationships between different concepts of approximate proper efficiency are stated.

Comparison of Some Scalarization Methods in Multiobjective Optimization

The paper presents an analysis, characterizations and comparison of six commonly used scalarization methods in multiobjective optimization. The properties of these methods are investigated with respect to the basic characteristics such as ordering cone, convexity and boundedness, the ability of generating proper efficient solutions, the ability to consider reference points which is a choice of decision maker as a solution and weighting preferences of decision maker, the number of additional constraints and decision variables. The paper also presents new characteristics for these methods and relations between them. The main characteristics of these scalarization methods are illustrated on the same example.

Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone

AbstractIn this work, we consider a multiobjective optimization problem in which the ordering cone is assumed to be polyhedral. In this framework, we characterize proper efficient solutions through nonlinear scalarization and a kind of polyhedral dilating cones. The main results are based on a characterization of weak efficient solutions, for which no convexity hypotheses are required. Moreover, the construction of these dilating cones allows us to obtain scalarization results that are easier to handle, and attractive from a computational point of view, since they are formulated in terms of a perturbation of the matrix that defines the ordering cone. Finally, when the feasible set is given by a cone constraint, we derive necessary and sufficient optimality conditions via a kind of scalar nonlinear Lagrangian.

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.

On connectedness of the set of efficient solutions for generalized systems

In this paper, we first give a density theorem. We will see that, under some suitable conditions, the set of positive proper efficient solutions is dense in the set of the efficient solutions. Finally, we discuss about the connectedness for the set of the efficient solutions of a generalized system.