Characterizations of generalized monotone maps

This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduced for maps. They were related to generalized convexity properties of functions in the case of gradient maps. In the present paper, we derive first-order characterizations of generalized monotone maps based on a geometrical analysis of generalized monotonicity. These conditions are both necessary and sufficient for generalized monotonicity. Specialized results are obtained for the affine case.     

Generalized Invexity and Generalized Invariant Monotonicity

In this paper, several kinds of invariant monotone maps and generalized invariant monotone maps are introduced. Some examples are given which show that invariant monotonicity and generalized invariant monotonicity are proper generalizations of monotonicity and generalized monotonicity. Relationships between generalized invariant monotonicity and generalized invexity are established. Our results are generalizations of those presented by Karamardian and Schaible.     

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.     

广义凸函数的特征性质

提出广义凸集、广义凸函数、中间点广义凸函数、端点广义凸函数四个定义,通过定义条件P1,研究条件P1所蕴含的等式关系,进而得到一个基础性定理一稠密性定理和一个相对条件较弱的推论,最后将结果应用于若干不同类型的广义凸函数类,尤其是s-凸函数、几何凸函数、rp-凸函数,得到它们所共有的一个特征性质,即满足稠密性定理．     

Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps

We establish connections between some concepts of generalized monotonicity for set-valued maps introduced earlier and some notions of generalized convexity. Moreover, a notion of pseudomonotonicity for set-valued maps is introduced; it is shown that, if a function f is continuous, then its pseudoconvexity is equivalent to the pseudomonotonicity of its generalized subdifferential in the sense of Clarke and Rockafellar.     

广义次似凸集值优化的鞍点定理

讨论广义次似凸集值优化的鞍点定理.给出广义次似凸集值映射的两个性质.定义广义次似凸集值优化的Fritz-John鞍点和Kuhn-Tucker鞍点.获得一系列广义次似凸集值优化的鞍点定理.     

Invariance properties of generalized monotonicity

Generalized monotone maps are studied under affine variable transformations. The results enable us to generate generalized monotone matrices of any size. Various necessary conditions and sufficient conditions for generalized monotone matrices are derived. Furthermore. admissible translations of generalized monotone linear maps are studied. Finally, the maximal domain of generalized monotonicity is characterized.     

Existence of Solutions and Variational Principles for Generalized Vector Systems

By means of generalized KKM theory, we prove a result on the existence of solutions and we establish general variational principles, that is, vector optimization formulations of set-valued maps for vector generalized systems. A perturbation function is involved in general variational principles. We extend the theory of gap functions for vector variational inequalities to vector generalized systems and we prove that the solution sets of the related vector optimization problems of set-valued maps contain the solution sets of vector generalized systems. A further vector optimization problem is defined in such a way that its solution set coincides with the solution set of a weak vector generalized system. Research carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2007. L.C. Ceng research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). J.C. Yao research was partially supported by the National Science Center for Theoretical Sciences at Tainan.     

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。     

Monotone Operators Representable by l.s.c. Convex Functions

A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone operators by convex functions. In the finite-dimensional case, we identify the class of monotone operators that admit a convex representation as the one consisting of intersections of maximal monotone operators and characterize the monotone operators that have a unique maximal monotone extension.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.