Approximate solutions and scalarization in set-valued optimization

Abstract

Certain notions of approximate weak efficient solutions are considered for a set-valued optimization problem based on vector and set criteria approaches. For approximate solutions based on the vector approach, a characterization is provided in terms of an extended Gerstewitz’s function. For the set approach case, two notions of approximate weak efficient solutions are introduced using a lower and an upper quasi order relations for sets and further compactness and stability aspects are discussed for these approximate solutions. Existence and scalarization using a generalized Gerstewitz’s function are also established for approximate solutions, based on the lower set order relation.     

New nonlinear scalarization functions and applications

In this paper, new nonlinear scalarization functions, which are different from the Gerstewitz function, are introduced. Some properties of these functions are discussed, and are used to prove new results on the existence of solutions of generalized vector quasi-equilibrium problems with moving cones and the lower semicontinuity of solution mappings of parametric vector quasi-equilibrium problems. Detailed comparisons between our results and those obtained by using the Gerstewitz function (for existence theorems) and by other approaches (for the case of solution stability) are given. Illustrating examples are provided.     

Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization

In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions.     

On Approximate Solutions in Vector Optimization Problems Via Scalarization

This work deals with approximate solutions in vector optimization problems. These solutions frequently appear when an iterative algorithm is used to solve a vector optimization problem. We consider a concept of approximate efficiency introduced by Kutateladze and widely used in the literature to study this kind of solutions. Necessary and sufficient conditions for Kutateladze’s approximate solutions are given through scalarization, in such a way that these points are approximate solutions for a scalar optimization problem. Necessary conditions are obtained by using gauge functionals while monotone functionals are considered to attain sufficient conditions. Two properties are then introduced to describe the idea of parametric representation of the approximate efficient set. Finally, through scalarization, characterizations and parametric representations for the set of approximate solutions in convex and nonconvex vector optimization problems are proved and the obtained results are applied to Pareto problems. AMS Classification:90C29, 49M37 This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.     

Construction of functions with uniform sublevel sets

In this paper, we study extended real-valued functions with uniform sublevel sets. The sublevel sets are defined by a linear shift of a set in a specified direction. We prove that the class of these functions coincides with the class of Gerstewitz functionals. In this way, we obtain a formula for the construction of such functions. The sublevel sets of Gerstewitz functionals are characterized and illustrated by examples. The results contain statements for translative functions, which are just the functions with uniform sublevel sets considered. The investigated functions are defined on an arbitrary real vector space without assuming any topology or convexity.     

Hölder continuity of the unique solution to parametric vector quasiequilibrium problems via nonlinear scalarization

In this paper, by virtue of the nonlinear scalarization function commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of the unique solution to a parametric vector quasiequilibrium problem is studied based on nonlinear scalarization approach, under three different kinds of monotonicity hypotheses. The globally Lipschitz property of the nonlinear scalarization function is fully employed. Our approach is totally different from the ones used in the literature, and our results not only generalize but also improve the corresponding ones in some related works.     

Cones Admitting Strictly Positive Functionals and Scalarization of Some Vector Optimization Problems

This paper is concerned with cones admitting strictly positive functionals and scalarization methods in multiobjective optimization. Assuming that the ordering cone admits strictly positive functionals or possesses a base in normed spaces or is a supernormal cone in a Banach space, we give scalar and scalar proper representations for vector optimization problems with convex and naturally quasiconvex data.     

向量优化中\varepsilon-真有效解的非线性标量化性质

利用G\"{o}pfert等提出的非线性标量化函数给出了向量优化中\varepsilon-真有效解的一个非线性标量化性质, 并提出几个例子对主要结果进行了解释.     

Vector optimization and generalized Lagrangian duality

In this paper, foundations of a new approach for solving vector optimization problems are introduced. Generalized Lagrangian duality, related for the first time with vector optimization, provides new scalarization techniques and allows for the generation of efficient solutions for problems which are not required to satisfy any convexity assumptions.     

New Scalarizing Approach to the Stability Analysis in Parametric Generalized Ky Fan Inequality Problems

This paper gives sufficient conditions for the upper and lower semicontinuities of the solution mapping of a parametric mixed generalized Ky Fan inequality problem. We use a new scalarizing approach quite different from traditional linear scalarization approaches which, in the framework of the stability analysis of solution mappings of equilibrium problems, were useful only for weak vector equilibrium problems and only under some convexity and strict monotonicity assumptions. The main tools of our approach are provided by two generalized versions of the nonlinear scalarization function of Gerstewitz. Our stability results are new and are obtained by a unified technique. An example is given to show that our results can be applied, while some corresponding earlier results cannot.     

向量优化中基于拟内部的弱C(\varepsilon)-有效解的标量化

首先获得了co-radiant集的一些拟内部性质. 进而在邻近C(\varepsilon)-次似凸性假设条件下, 建立了相应的择一性定理, 并给出了基于拟内部的集值向量优化问题弱C(\varepsilon)-有效解的线性标量化结果.     

Improvement sets and vector optimization

In this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems, where the preference relation is defined via an improvement set E. To be precise, we extend the notion of E-optimal point due to Chicco et al. in [4] to a general (non-necessarily Pareto) quasi ordered linear space and we study its properties. In particular, we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions. Moreover, we obtain necessary and sufficient conditions for E-optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general (nonconvex) case. By applying the obtained results to certain improvement sets we generalize well-known results of the literature referred to efficient, weak efficient and approximate efficient solutions of vector optimization problems.     

集值映射向量优化问题的ε-真有效解

本文讨论集值映射向量优化问题的ε-真有效解。在集值映射为广义锥-次类凸的假设下,建立了这种解的标量化定理,ε-Lagrange乘子定理,ε-真鞍点定理和ε-真对偶性定理。     

Characterization of properly optimal elements with variable ordering structures

In vector optimization with a variable ordering structure, the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. In recent publications, it was started to develop a comprehensive theory for these vector optimization problems. Thereby, also notions of proper efficiency were generalized to variable ordering structures. In this paper, we study the relation between several types of proper optimality. We give scalarization results based on new functionals defined by elements from the dual cones which allow complete characterizations also in the nonconvex case.     

A Nonlinear Scalarization Function and Generalized Quasi-vector Equilibrium Problems

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. In this paper we introduce a nonlinear scalarization function for a variable domination structure. Several important properties, such as subadditiveness and continuity, of this nonlinear scalarization function are established. This nonlinear scalarization function is applied to study the existence of solutions for generalized quasi-vector equilibrium problems. This paper is dedicated to Professor Franco Giannessi for his 68th birthday     

On quadratic scalarization of vector optimization problems in Banach spaces

We study vector optimization problems in partially ordered Banach spaces and suppose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We discuss the so-called adaptive scalarization of such problems and show that the corresponding scalar non-linear optimization problems can be by-turn approximated by quadratic minimization problems.     

Characterization of Approximate Solutions of Vector Optimization Problems with a Variable Order Structure

In this paper, we deal with approximate solutions in vector-optimization problems with respect to a variable order structure. In the case of exact solutions of a vector optimization problem, especially in the variable order case, authors use a cone or a pointed convex cone-valued map in order to describe the solution concepts but in this paper, we use a set-valued map and this map is not a (pointed convex) cone-valued map necessarily. We characterize these solution concepts by a general scalarization method by means of nonlinear functionals. In the last section, an extension of Ekeland’s variational principle for a vector optimization problem with a variable order structure is given.     

Vector network equilibrium problems and nonlinear scalarization methods

The conventional equilibrium problem found in many economics and network models is based on a scalar cost, or a single objective. Recently, equilibrium problems based on a vector cost, or multicriteria, have received considerable attention. In this paper, we study a scalarization method for analyzing network equilibrium problems with vector-valued cost function. The method is based on a strictly monotone function originally proposed by Gerstewitz. Conditions that are both necessary and sufficient for weak vector equilibrium are derived, with the prominent feature that no convexity assumptions are needed, in contrast to other existing scalarization methods.     

Efficiency and Henig Efficiency for Vector Equilibrium Problems

We introduce the concept of Henig efficiency for vector equilibrium problems, and extend scalarization results from vector optimization problems to vector equilibrium problems. Using these scalarization results, we discuss the existence of the efficient solutions and the connectedness of the set of Henig efficient solutions to the vector-valued Hartman–Stampacchia variational inequality.     

Scalarization and optimality conditions for vector equilibrium problems

In this paper, we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions, Henig efficient solutions, and globally efficient solutions to the vector equilibrium problems without the convexity assumption. Using nonsmooth analysis and the scalarization results, we provided the necessary conditions for weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems. By the assumption of convexity, we gave sufficient conditions for those solutions. As applications, we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems.