Benson proper efficiency for vector optimization of generalized subconvexlike set-valued maps in ordered linear spaces

Several equivalent statements of generalized subconvexlike set-valued map are established in ordered linear spaces. Using vector closure, we introduce Benson proper efficient solution of vector optimization problem. Under the assumption of generalized subconvexlikeness, scalarization, multiplier and saddle point theorems are obtained in the sense of Benson proper efficiency.

THE OPTIMALITY CONDITIONS OF NONCONVEXSET-VALUED VECTOR OPTIMIZATION

The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz John optimality condition of set-valued vector optimization is established. Finally, under the assumption of pseudoconvexity, the optimality condition is proved to be sufficient.

Optimality for Henig Proper Efficiency in Vector Optimization Involving Dini Set-Valued Directional Derivatives

This note studies the optimality conditions of vector optimization problems involving generalized convexity in locally convex spaces. Based upon the concept of Dini set-valued directional derivatives, the necessary and sufficient optimality conditions are established for Henig proper and strong minimal solutions respectively in generalized preinvex vector optimization problems.

Gap Functions and Error Bounds for Set-Valued Vector Quasi Variational Inequality Problems

One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of gap function. The gap functions are useful tools in deriving the error bounds which provide an estimated distance between a specific point and the exact solution of variational inequality problem. In this paper, we follow a similar approach for set-valued vector quasi variational inequality problems and define the gap functions based on scalarization scheme as well as the one with no scalar parameter. The error bounds results are obtained under fixed point symmetric and locally α-Holder assumptions on the set-valued map describing the domain of solution space of a set-valued vector quasi variational inequality problem.

Optimality conditions in set optimization employing higher-order radial derivatives

There are two approaches of defining the solutions of a set-valued optimization problem： vector criterion and set criterion. This note is devoted to higher-order optimality conditions using both criteria of solutions for a constrained set-valued optimization problem in terms of higher-order radial derivatives. In the case of vector criterion, some optimality conditions are derived for isolated （weak） minimizers. With set criterion, necessary and sufficient optimality conditions are established for minimal solutions relative to lower set-order relation.

Semicontinuity of the Minimal Solution Set Mappings for Parametric Set-Valued Vector Optimization Problems

With the help of a level mapping,this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems.First,we introduce a kind of level mapping which generalizes one given in Han and Gong(Optimization 65:1337–1347,2016).Then,we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping.Finally,in terms of the semicontinuity of the level mapping,we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric setvalued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.

Optimality Conditions of the Set-valued Optimization Problem with Generalized Cone Convex Set-valued Maps Characterized by Contingent Epiderivative

In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.

Henig Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions

In this paper, we study Henig efficiency in vector optimization with nearly cone-subconvexlike set-valued function. The existence of Henig efficient point is proved and characterization of Henig efficiency is established using the method of Lagrangian multiplier. As an interesting application of the results in this paper, we establish a Lagrange multiplier theorem for super efficiency in vector optimization with nearly conesubconvexlike set-valued function.

Upper and Lower Semicontinuity of Solution Sets for Parametric Generalized Vector Quasi-equilibrium Problems

In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters are investigated.

集值映射的Michel-Penot方向导数与Michel-Penot次微分

采用标量化方法,研究集值映射的方向导数和次微分.借助Gerstewizt函数,给出了集值映射的Michel-Penot方向导数和Michel-Penot次微分的概念,研究了它们的性质,建立了它们的计算法则,并利用Michel-Penot次微分给出了无约束集值优化问题的优化条件.

Second-order Optimality Conditions for Cone-subarcwise Connected Set-valued Optimization Problems

The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set,respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems.

NONEMPTY INTERSECTION THEOREMS AND SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS IN PRODUCT G-CONVEX SPACES

By using an existence theorems of maximal elements for a family of set-valued mappings in G-convex spaces due to the author, some new nonempty intersection theorems for a family of set-valued mappings were established in noncompact product G-convex spaces. As applications, some equilibrium existence theorems for a system of generalized vector equilibrium problems were proved in noncompact product G-convex spaces. These theorems unify, improve and generalize some important known results in literature.

NONEMPTY INTERSECTION THEOREMS AND SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS IN PRODUCT G-CONVEX SPACES

By using an existence theorems of maximal elements for a family of set-valued mappings in G-convex spaces due to the author, some new nonempty intersection theorems for a family of set-valued mappings were established in noncompact product G-convex spaces. As applications, some equilibrium existence theorems for a system of generalized vector equilibrium problems were proved in noncompact product G-convex spaces. These theorems unify, improve and generalize some important known results in literature.

LEVITIN-POYAK WELL-POSEDNESS OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS WITH FUNCTIONAL CONSTRAINTS

In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given.

On Ekeland's Variational Principle for Set-Valued Mappings

In this paper, we derive a general vector Ekeland variational principle for set-valued mappings, which has a dosed relation to εk^0 -efficient points of set-valued optimization problems. The main result presented in this paper is a generalization of the corresponding result in [3].

向量集值优化问题f-有效解的最优性条件

利用分离定理并借助集值映射的余切上图导数、Y-上图导数、径向上图导数和Clarke正切上图导数给出了向量集值优化问题,取得f-有效解的4个最优性条件。

向量集值优化问题全局真有效解的最优性条件

利用分离定理并借助于集值映射的余切上图导数、Y-上图导数和Clarke正切上图导数,给出了向量集值优化问题取得全局真有效解的3个最优性条件.

拟凸向量值映射次微分性质及优化问题的最优性条件

本文研究了拟凸向量值映射的次微分及其拟凸向量优化问题的最优性条件.首先,引进恰当K-拟凸的概念,并利用△函数对其进行标量化,得到恰当K-拟凸的等价刻画.然后,给出拟凸向量值映射的四种次微分的定义,并研究了它们的性质.最后,利用拟凸向量值映射的次微分研究拟凸向量优化问题弱有效解的最优性条件,并用例子说明其合理性.

A Revised Pre-Order Principle and Set-Valued Ekeland Variational Principles with Generalized Distances

In my former paper ＂A pre-order principle and set-valued Ekeland variational principle＂ （see [J. Math. Anal. Applo, 419, 904 937 （2014）]）, we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles （denoted by EVPs） in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy＇s EVP in [On generalized Ekeland＇s variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381-396 （2011）], where the perturbation contains a weak T-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy＇s EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.

集值映射向量优化问题的强有效性

引进并较为系统地研究集值映射向量优化问题的强有效性，获得了包括标量化、Ｌａｇｒａｎｇｅ乘子。