Existence of weakly efficient solutions in nonsmooth vector optimization

In this paper we study the existence of weakly efficient solutions for some nonsmooth and nonconvex vector optimization problems. We consider problems whose objective functions are defined between infinite and finite-dimensional Banach spaces. Our results are stated under hypotheses of generalized convexity and make use of variational-like inequalities.     

Existence and continuity of solutions for vector optimization

Conditions are obtained for the existence of a weak minimum for constrained vector optimization, with a less restrictive compactness hypothesis than usual. Conditions are also derived for the upper and lower semicontinuity of the multifunction describing weak minimum points. The results are applicable to semi-infinite vector optimization.     

Optimality conditions for proper efficient solutions of vector set-valued optimization

Based on the concept of an epiderivative for a set-valued map introduced in J. Nanchang Univ. 25 (2001) 122-130, in this paper, we present a few necessary and sufficient conditions for a Henig efficient solution, a globally proper efficient solution, a positive properly efficient solution, an f-efficient solution and a strongly efficient solution, respectively, to a vector set-valued optimization problem with constraints.     

Existence of efficient solutions for vector maximization problems

The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points. Examples are provided to illustrate these results.The author would like to thank Professor T. L. Morin for his helpful comments. Thanks also go to an anonymous reviewer for his useful comments concerning an earlier version of this paper.The author would like to acknowledge a useful discussion with Professor G. Bitran which helped in motivating Example 4.1.     

Efficient solutions in generalized linear vector optimization

This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized linear vector optimization problems is obtained. We also prove that the efficient solution set of a generalized linear vector optimization problem in a locally convex Hausdorff topological vector space is the union of finitely many generalized polyhedral convex sets and it is connected by line segments.     

Nonconvex separation theorems and some applications in vector optimization

Separation theorems for an arbitrary set and a not necessarily convex set in a linear topological space are proved and applied to vector optimization. Scalarization results for weakly efficient points and properly efficient points are deduced.     

On vector optimization problems and vector variational inequalities using convexificators

In this paper, we establish some results which exhibit an application of convexificators in vector optimization problems (VOPs) and vector variational inequaities involving locally Lipschitz functions. We formulate vector variational inequalities of Stampacchia and Minty type in terms of convexificators and use these vector variational inequalities as a tool to find out necessary and sufficient conditions for a point to be a vector minimal point of the VOP. We also consider the corresponding weak versions of the vector variational inequalities and establish several results to find out weak vector minimal points.     

Complete efficiency and interdependencies between objective functions in vector optimization

Summary Vector optimization problems in linear spaces with respect to general domination sets are investigated including corollaries to Pareto-optimality and weak efficiency. The results contain equivalences between vector optimization problems, interdependencies between objective functions and domination sets, statements about the influence of perturbed objective functions on the decision maker's preferences and thus on the domination set, comparisons of efficiency with respect to polyhedral cones with Pareto-optimality, changes in the objective functions which restrict, extend or do not alter the set of Pareto-optima, possibilities for the use of domination sets immediately in the decision space. Conditions for complete efficiency are given.

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Zusammenfassung Untersucht werden Vektoroptimierungsprobleme in linearen Räumen bezüglich allgemeiner Dominanzmengen einschließlich Folgerungen für Pareto-Optimalität und schwache Effizienz. Die Ergebnisse enthalten Äquivalenzen zwischen Vektoroptimierungsproblemen, Wechselwirkungen zwischen Zielfunktionen und Dominanzmengen, Aussagen über den Einfluß gestörter Zielfunktionen auf die Präferenzen des Entscheidungsträgers und somit auf die Dominanzmenge, Vergleiche von Effizienz bezüglich polyedrischer Kegel mit Pareto-Optimalität, Änderungen in den Zielfunktionen, die die Menge der Pareto-Optima einschränken, erweitern oder nicht beeinflussen, Möglichkeiten für die Nutzung von Dominanzmengen unmittelbar im Entscheidungsraum. Bedingungen für vollständige Effizienz werden gegeben.

     

Geometrical characterization of weakly efficient points

In this note, we present a geometrical characterization of the set of weakly efficient points in constrained convex multiobjective optimization problems, valid for a compact set of objectives.     

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.     

Parametric vector optimization

In this paper we consider a sequence of vector optimization problems. We aim to generalize a vector condition that relates the parametric function and the limit function. In particular, we recover our condition given in the scalar case. Our stability approach is such that the limit of the sequence of solutions that correspond to vector optimization problems to be a solution of a limit vector optimization problem. Therefore, one can view our statement as an existence result. This general framework has been used in several previous works. In our main theorem, we use the notion of strong lower cone-semi-continuity. An example is given to illustrate why only cone-lower semi-continuity for the limit function is not sufficient for our result.     

Existence of solutions for vector optimization

In this paper, we prove the existence of a weak minimum for constrained vector optimization problem by making use of vector variational-like inequality and preinvex functions.     

On Minty variational principle for nonsmooth vector optimization problems with generalized approximate convexity

In this paper, we consider a nonsmooth vector optimization problem involving locally Lipschitz generalized approximate convex functions and find some relations between approximate convexity and generalized approximate convexity. We establish relationships between vector variational inequalities and nonsmooth vector optimization problem using the generalized approximate convexity as a tool.     

A unified vector optimization problem: complete scalarizations and applications

The aim of this article is to introduce and analyse a general vector optimization problem in a unified framework. Using a well-known nonlinear scalarizing function defined by a solid set, we present complete scalarizations of the solution set to the vector problem without any convexity assumptions. As applications of our results we obtain new optimality conditions for several classical optimization problems by characterizing their solution set.     

Efficient and weakly efficient solutions for the vector topical optimization

In this paper, we consider some scalarization functions, which consist of the generalized min-type function, the so-called plus-Minkowski function and their convex combinations. We investigate the abstract convexity properties of these scalarization functions and use them to identify the maximal points of a set in an ordered vector space. Then, we establish some versions of Farkas type results for the infinite inequality system involving vector topical functions. As applications, we obtain the necessary and sufficient conditions of efficient solutions and weakly efficient solutions for a vector topical optimization problem, respectively.     

Planar point-objective location problems with nonconvex constraints: A geometrical construction

The planar point-objective location problem has attracted considerable interest among Location Theory researchers. The result has been a number of papers giving properties or algorithms for particular instances of the problem. However, most of these results are only valid when the feasible region where the facility is to be located is the whole space ², which is a rather inaccurate approximation in many real world location problems.In this paper, the feasible region is allowed to be any closed, not necessarily convex, setS in ². The special structure of this nonconvex vector-optimization problem is exploited, leading to a geometrical resolution procedure when the feasible regionS can be decomposed into a finite number of (not necessarily disjoint) polyhedra.     

Duality and saddle-points for convex-like vector optimization problems on real linear spaces

Usually, finite dimensional linear spaces, locally convex topological linear spaces or normed spaces are the framework for vector and multiojective optimization problems. Likewise, several generalizations of convexity are used in order to obtain new results. In this paper we show several Lagrangian type duality theorems and saddle-points theorems. From these, we obtain some characterizations of several efficient solutions of vector optimization problems (VOP), such as weak and proper efficient solutions in Benson’s sense. These theorems are generalizations of preceding results in two ways. Firstly, because we consider real linear spaces without any particular topology, and secondly because we work with a recently appeared convexlike type of convexity. This new type, designated GVCL in this paper, is based on a new algebraic closure which we named vector closure. This research for the second author was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.     

集值优化强有效解的广义二阶锥方向导数刻画

在实赋范线性空间中考虑集值优化问题的强有效性．借助Henig扩张锥和基泛函的性质,利用广义二阶锥方向相依导数,得到受约束于集值映射的优化问题,取得强有效元的二阶最优性必要条件．当目标函数为近似锥一次类凸映射时,利用强有效点的标量化定理,得到集值优化问题,取得强有效元的二阶充分条件．