Some relations between ε-directional derivative and ε-generalized weak subdifferential

Document Type: Research Paper

Authors

Shahid Bahonar university of Kerman

Abstract

In this paper, we study ε-generalized weak subdifferential for vector valued functions defined on a real ordered topological vector space X. We give various characterizations of ε-generalized weak subdifferential for this class of functions. It is well known that if the function f : X R is subdifferentiable at x0 X, then f has a global minimizer at x0 if and only if 0 f(x0). We show that a similar result can be obtained for ε-generalized weak subdifferential. Finally, we investigate some relations between ε-directional derivative and ε-generalized weak subdifferential. In fact, in the classical subdifferential theory, it is well known that if the function f : X R is subdifferentiable at x0 X and it has directional derivative at x0 in the direction u X, then the relation f (x0, u) ≥ ⟨u, x, xf(x0) is satisfied. We prove that a similar result can be obtained for ε- generalized weak subdifferential.

Keywords


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