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Acta Mathematica Vietnamica

SOME CONDITIONS FOR NONEMPTINESS OF γ-SUBDIFFERENTIALS OF γ-CONVEX FUNCTIONS

icon-email NGUYEN NGOC HAI

Abstract

γ-subdifferential is a concept which can be used for global optimization. If x is a global minimizer of an arbitrary function f$$ then 0γf(x), where γf(x) is the γ-subdifferential of f at x. In particular, γf(x) at a global minimizer x. In this paper we investigate the nonemptiness and the monotonicity of γ-subdifferentials of γ-convex functions. Some sufficient conditions are stated for the nonemptiness of the γ-subdifferential of a symmetrically γ-convex function at a point. It is proved that for a symmetrically γ-convex function, the G\^{a}teaux derivative (when it exists) at a point belongs to the γ-subdifferential at that point. A relation between the γ-subdifferential and the Clarke generalized gradient of a symmetrically γ-convex function is also presented.