SOME CONDITIONS FOR NONEMPTINESS OF -SUBDIFFERENTIALS OF -CONVEX FUNCTIONS
NGUYEN NGOC HAI
Abstract
-subdifferential is a concept which can be used for global optimization. If is a global minimizer of an arbitrary function f$$ then , where is the -subdifferential of at . In particular, at a global minimizer . In this paper we investigate the nonemptiness and the monotonicity of -subdifferentials of -convex functions. Some sufficient conditions are stated for the nonemptiness of the -subdifferential of a symmetrically -convex function at a point. It is proved that for a symmetrically -convex function, the G\^{a}teaux derivative (when it exists) at a point belongs to the -subdifferential at that point. A relation between the -subdifferential and the Clarke generalized gradient of a symmetrically -convex function is also presented.