Well-Posedness and Scalarization in Vector Optimization

In this paper, we study several existing notions of well- posedness for vector optimization problems. We separate them into two classes and we establish the hierarchical structure of their relationships. Moreover, we relate vector well-posedness and well-posedness of an appropriate scalarization. This approach allows us to show that, under some compactness assumption, quasiconvex problems are well posed.The authors thank Professor C. Zălinescu for pointing out some inaccuracies in Ref. 11. His remarks allowed the authors to improve the present work.     

向量值优化问题的适定性

将向量值优化问题的适定性理论推广到具有W-距离的拟度量空间中,给出B-适定性及DH-适定性的概念,并得到相应的判别准则．定义一类非线性标量化函数,利用这类标量化函数的性质将向量值优化问题转化为标量值优化问题,得到原问题与标量化问题的适定性之间的等价关系．     

Scalarization and Nonlinear Scalar Duality for Vector Optimization with Preferences that are not necessarily a Pre-order Relation

We consider problems of vector optimization with preferences that are not necessarily a pre-order relation. We introduce the class of functions which can serve for a scalarization of these problems and consider a scalar duality based on recently developed methods for non-linear penalization scalar problems with a single constraint.     

广义锥-s次类凸向量优化的标量化和鞍点定理

讨论序拓扑向量空间中的约束向量优化问题.在广义锥-s次类凸假设下,得到了向量优化问题关于δ-弱有效解的标量化定理和Lagrange泛函的鞍点定理.     

可变序结构下向量优化中的一个新非线性标量化函数及其应用

在具有可变序结构的一般拓扑向量空间中定义了一个新的非线性标量化函数,讨论了该函数的主要性质.同时作为应用,通过该函数构造出了一族半范数和一类赋范线性空间,并在最后建立了该非线性标量化函数和半范数的上、下半连续性结论.     

Scalarization and Kuhn-Tucker-Like Conditions for Weak Vector Generalized Quasivariational Inequalities

We consider a weak vector generalized quasivariational inequality. By introducing a method of scalarization which does not require any assumption on the data and by using previous results of the authors concerning scalar generalized quasivariational inequalities, we present Kuhn-Tucker-like conditions for this problem in the case in which the set-valued operator of the constraints is defined by a finite number of inequalities     

A Scalarization Approach for Vector Variational Inequalities with Applications

We consider an approach to convert vector variational inequalities into an equivalent scalar variational inequality problem with a set-valued cost mapping. Being based on this property, we give an equivalence result between weak and strong solutions of set-valued vector variational inequalities and suggest a new gap function for vector variational inequalities. Additional examples of applications in vector optimization, vector network equilibrium and vector migration equilibrium problems are also given Mathematics Subject Classification(2000). 49J40, 65K10, 90C29     

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.     

Scalarization of Henig Proper Efficient Points in a Normed Space

In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base.     

非凸多目标优化中真有效解的一个非线性标量化性质

基于Pascoletti-Serafini标量化方法,利用罚函数思想提出了一类新的标量化函数,进而获得非凸多目标优化问题真有效解的充分条件和必要条件.该结果的建立不需要目标函数的像集有界这一条件,故文章是对Akbari等人[J.Optim.Theory Appl.,2018,178(2):560-590]建立的相应标量...     

混合向量变分不等式标量化及间隙函数误差界

利用Konnov对变分不等式问题的标量化方法,对一般的强变分不等式（SVI）和弱变分不等式（WVI）进行了进一步的推广.主要介绍了基于集值映射的强广义混合向量变分不等式（SGMVVI）和弱广义混合向量变分不等式（WGMVVI）,考虑了与它们相关的间隙函数,在合适的条件下讨论了强广义混合集值变分不等式（SGMVI）的间隙函数和SGMVVI的间隙函数之间的关系,以及WGMVVI和SGMVI的间隙函数之间的关系,最后讨论了它们的间隙函数的全局误差界.     

带变动序结构的向量优化问题的E-最优元

推广固定锥序下的改进集概念到变动序关系.引入了带变动序结构的向量优化问题的E-最优元.应用Tammer-Weidner意义下的非线性标量化函数,给出了向量优化问题E-最优元的标量化刻画,建立了带变动序结构的向量优化问题的E-最优元的必要和充分最优性条件.     

Scalarization and decomposition of vector variational inequalities governed by bifunctions

In this article we study the structure of solution sets within a special class of generalized Stampacchia-type vector variational inequalities, defined by means of a bifunction which takes values in a partially ordered Euclidean space. It is shown that, similar to multicriteria optimization problems, under appropriate convexity assumptions, the (weak) solutions of these vector variational inequalities can be recovered by solving a family of weighted scalar variational inequalities. Consequently, it is deduced that the set of weak solutions can be decomposed into the union of the sets of strong solutions of all variational inequalities obtained from the original one by selecting certain components of the bifunction which governs it.     

Characterizations of Variable Domination Structures via Nonlinear Scalarization

In this paper, a nonlinear scalarization function is introduced for a variable domination structure. It is shown that this function is positively homogeneous, subadditive, and strictly monotone. This nonlinear function is then applied to characterize the weakly nondominated solution of multicriteria decision making problems and the solution of vector variational inequalities.     

Pointwise Well-Posedness of Perturbed Vector Optimization Problems in a Vector-Valued Variational Principle

In this note, we point out and correct some errors in Ref. 1. Another type of pointwise well-posedness and strong pointwise well-posedness of vector optimization problems is introduced. Sufficient conditions to guarantee this type of well-posedness are provided for perturbed vector optimization problems in connection with the vector-valued Ekeland variational principle.     

Scalarization and lagrange duality in multiobjective optimization

In this paper, we are concerned with scalarization and the Lagrange duality in multiobjective optimization. After exposing a property of a cone-subconvexlike function, we prove two theorems on scalarization and three theorems of the Lagrange duality.     

Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions

A new class of generalized convex set-valued functions, termed nearly-subconvexlike functions, is introduced. This class is a generalization of cone-subconvexlike maps, nearly-convexlike set-valued functions, and preinvex set-valued functions. Properties for the nearly-subconvexlike functions are derived and a theorem of the alternative is proved. A Lagrangian multiplier theorem is established and two scalarization theorems are obtained for vector optimization.     

Extended Well-Posedness Properties of Vector Optimization Problems

In this paper, the concept of extended well-posedness of scalar optimization problems introduced by Zolezzi is generalized to vector optimization problems in three ways: weakly extended well-posedness, extended well-posedness, and strongly extended well-posedness. Criteria and characterizations of the three types of extended well-posedness are established, generalizing most of the results obtained by Zolezzi for scalar optimization problems. Finally, a stronger vector variational principle and Palais-Smale type conditions are used to derive sufficient conditions for the three types of extended well-posedness.     

生成锥内部凸-锥-类凸集值优化问题的标量化和鞍点

讨论集值向量优化的标量化和鞍点问题.在生成锥内部凸-锥-类凸假设下,建立了集值向量优化问题在（弱）有效和Benson真有效意义下的标量化定理和鞍点定理.