广义单调性与广义凸性

Generalized monotonicity and generalized convexity

在广义单调性方面做进一步的推广,并建立函数的广义凸性与其梯度向量的广义单调性之间的等价关系.首先,建立F单调映射、F伪单调映射和F拟单调映射概念.其次,利用可微F凸(伪凸、拟凸)函数的梯度等价刻画,结合广义单调映射概念以及微分性质,研究广义凸性与广义单调性的内在联系:在一定条件下,f是K上的F凸函数,当且仅当?f是K上的F单调映射;f是K上的F伪凸函数,当且仅当?f是K上的F伪单调映射;f是K上的F拟凸函数,当且仅当?f是K上的F拟单调映射.最后,给出某些具体广义凸函数的广义单调性刻画.

This article extends some conclusions of generalized monotonicity and establishes equivalent relationship between generalized convexity of functions and the generalized monotonicity of their gradient. Firstly,give the concepts of F-monotone maps,F-pseudo monotone maps and F-quasi monotone maps. Secondly,use gradient equivalent descriptions of differentiable F-convex(pseudo convex,quasi-convex) functions,and combine the concepts of generalized monotone maps and the properties of differentiation to study the relationship between generalized monotonicity and generalized convexity. The results show that under certain conditions,f is F-convex functions over K,if and only if?f is F-monotone maps over K. f is F-pseudo functions over K,if and only if ?f is F-pseudo monotone maps over K. f is F-quasi functions over K,if and only if ?f is F-quasi monotone maps over K. Lastly,based on the application,give some generalized monotonicity descriptions of generalized convex functions.