On Generalized Vector Equilibrium Problems

A new generalized vector equilibrium problem involving set-valued mappings and the proper quasi concavity of set-valued mappings in topological vector spaces are introduced; its existence theorems and the convexity of the solution sets are established.     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

Monotone and Accretive Vector Fields on Riemannian Manifolds

The relationship between monotonicity and accretivity on Riemannian manifolds is studied in this paper and both concepts are proved to be equivalent in Hadamard manifolds. As a consequence an iterative method is obtained for approximating singularities of Lipschitz continuous, strongly monotone mappings. We also establish the equivalence between the strong convexity of functions and the strong monotonicity of its subdifferentials on Riemannian manifolds. These results are then applied to solve the minimization of convex functions on Riemannian manifolds.     

Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions

In this paper, by using the scalarization approach of Konnov, several kinds of strong and weak scalar variational inequalities (SVI and WVI) are introduced for studying strong and weak vector variational inequalities (SVVI and WVVI) with set-valued mappings, and their gap functions are suggested. The equivalence among SVVI, WVVI, SVI, WVI is then established under suitable conditions and the relations among their gap functions are analyzed. These results are finally applied to the error bounds for gap functions. Some existence theorems of global error bounds for gap functions are obtained under strong monotonicity and several characterizations of global (respectively local) error bounds for the gap functions are derived.     

Convexity with respect to a differential equation

The concepts of convexity of a set, convexity of a function and monotonicity of an operator with respect to a second-order ordinary differential equation are introduced in this paper. Several well-known properties of usual convexity are derived in this context, in particular, a characterization of convexity of function and monotonicity of an operator. A sufficient optimality condition for a optimization problem is obtained as an application. A number of examples of convex sets, convex functions and monotone operators with respect to a differential equation are presented.     

Equilibrium Problems under Generalized Convexity and Generalized Monotonicity

Generalized convex functions preserve many valuable properties of mathematical programming problems with convex functions. Generalized monotone maps allow for an extension of existence results for variational inequality problems with monotone maps. Both models are special realizations of an abstract equilibrium problem with numerous applications, especially in equilibrium analysis (e.g., Blum and Oettli, 1994). We survey existence results for equilibrium problems obtained under generalized convexity and generalized monotonicity. We consider both the scalar and the vector case. Finally existence results for a system of vector equilibrium problems under generalized convexity are surveyed which have applications to a system of vector variational inequality problems. Throughout the survey we demonstrate that the results can be obtained without the rigid assumptions of convexity and monotonicity.     

On strongly α-preinvex functions

In this paper, by means of a series of counterexamples, we study in a systematic way the relationships among (pseudo, quasi) α-preinvexity, (strict, strong, pseudo, quasi) α-invexity and (strict, strong, pseudo, quasi) αη-monotonicity. Results obtained in this paper can be viewed as a refinement and improvement of the results of Noor and Noor [M.A. Noor, K.I. Noor, Some characterizations of strongly preinvex functions, J. Math. Anal. Appl. 316 (2006) 697–706].     

与GA-凸函数有关的两个函数

引入两个与GA-凸函数有关的函数,研究了它们的凸性、单调性及相互关系,获得了已有文献关于GA-凸函数的Hadamard型不等式的加细,也得到一些新的不等式.     

广义强向量拟均衡问题组解的存在性

本文研究了一类集值广义强向量拟均衡问题组解的存在性问题.利用集值映射的自然拟C-凸性和集值映射的下(-C)-连续性的定义和Kakutani-Fan-Glicksberg不动点定理,在不要求锥C的对偶锥C~*具有弱*紧基的情况下,建立了该类集值广义强向量拟均衡问题组解的存在性定理.所得结果推广了该领域的相关结果.     

Generalized minimax inequalities for set-valued mappings

In this paper, we study generalized minimax inequalities in a Hausdorff topological vector space, in which the minimization and the maximization of a two-variable set-valued mapping are alternatively taken in the sense of vector optimization. We establish two types of minimax inequalities by employing a nonlinear scalarization function and its strict monotonicity property. Our results are obtained under weaker convexity assumptions than those existing in the literature. Several examples are given to illustrate our results.     

Fuzzy Integrals of Set-vatued Function

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Vector Equilibrium Problems. Existence Theorems and Convexity of Solution Set

The natural quasi-concavity of set-valued mappings in an ordered vector space is introduced. Existence theorems for vector equilibrium problems involving set-valued monotone mappings are obtained and the convexity of the solution set is shown.     

Further on Set-Valued Equilibrium Problems and Applications to Browder Variational Inclusions

In this paper, we introduce some concepts of convexity and semicontinuity for real set-valued mappings similar to those of real single-valued mappings. Then, we obtain different results on the existence of solutions of set-valued equilibrium problems generalizing in a common way several old ones for both single-valued and set-valued equilibrium problems. Applications to Browder variational inclusions, with weakened conditions on the involved set-valued operator, are given.     

Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions

We derive a uniform (strong) Law of Large Numbers (LLN) for random set-valued mappings. The result can be viewed as an extension of both, a uniform LLN for random functions and LLN for random sets. We apply the established results to a consistency analysis of stationary points of sample average approximations of nonsmooth stochastic programs.     

Subgradient of distance functions with applications to Lipschitzian stability

The paper is devoted to studying generalized differential properties of distance functions that play a remarkable role in variational analysis, optimization, and their applications. The main object under consideration is the distance function of two variables in Banach spaces that signifies the distance from a point to a moving set. We derive various relationships between Fréchet-type subgradients and limiting (basic and singular) subgradients of this distance function and corresponding generalized normals to sets and coderivatives of set-valued mappings. These relationships are essentially different depending on whether or not the reference point belongs to the graph of the involved set-valued mapping. Our major results are new even for subdifferentiation of the standard distance function signifying the distance between a point and a fixed set in finite-dimensional spaces. The subdifferential results obtained are applied to deriving efficient dual-space conditions for the local Lipschitz continuity of distance functions generated by set-valued mappings, in particular, by those arising in parametric constrained optimization. Dedicated to Terry Rockafellar in honor of his 70th birthday. This research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451158.     

不可微函数的不变伪单调性与伪不变凸性

本文讨论不可微函数的强伪不变凸性和不变伪单调性之间的关系,还讨论了不可微函数的伪不变凸性与半严格预拟不变凸性.此外,还给出了一个集值映射的不变伪单调的等价定义.     

Merit functions and error bounds for constrained mixed set-valued variational inequalities via generalized f-projection operators

In this paper, we introduce and investigate a constrained mixed set-valued variational inequality (MSVI) in Hilbert spaces. We prove the solution set of the constrained MSVI is a singleton under strict monotonicity. We also propose four merit functions for the constrained MSVI, that is, the natural residual, gap function, regularized gap function and D-gap function. We further use these functions to obtain error bounds, i.e. upper estimates for the distance to solutions of the constrained MSVI under strong monotonicity and Lipschitz continuity. The approach exploited in this paper is based on the generalized f-projection operator due to Wu and Huang, but not the well-known proximal mapping.     

Perturbation approach to generalized Nash equilibrium problems with shared constraints

This paper aims to consider an application of conjugate duality in convex optimization to the generalized Nash equilibrium problems with shared constraints and nonsmooth cost functions. Sufficient optimality conditions for the problems regarding to players are rewritten as inclusion problems and the maximal monotonicity of set-valued mappings generated by the subdifferentials of functions from data of GNEP is proved. Moreover, some assertions dealing with solutions to GNEP are obtained and applications of splitting algorithms to the oligopolistic market equilibrium models are presented.     

向量优化中广义增广拉格朗日对偶理论及应用

作者介绍了一种基于向量值延拓函数的广义增广拉格朗日函数,建立了基于广义增广拉格朗日函数的集值广义增广拉格朗日对偶映射和相应的对偶问题,得到了相应的强对偶和弱对偶结果,将所获结果应用到约束向量优化问题.该文的结果推广了一些已有的结论.     

Existence of Solutions of New Generalized Mixed Vector Variational-Like Inequalities in Reflexive Banach Spaces

In this paper, we extend the concept of monotonicity for a vector set-valued mapping to semimonotonicity for a vector set-valued mapping. Then, we prove solvability results for a class of new generalized mixed vector variational-like inequalities by applying the Fan-KKM theorem and Nadler’s result. On the other hand, we introduce the concepts of complete semicontinuity and strong semicontinuity for vector multivalued mappings. Moreover, by using the Brouwer fixed point theorem, we prove the solvability for the class of generalized vector variational-like inequalities without monotonicity assumption. Using this result, we obtain a theorem and corollary that improve and extend some known results.