Mohebi, A., Mohebi, H. (2015). Some relations between ε-directional derivative and ε-generalized weak subdifferential. Wavelet and Linear Algebra, 2(1), 65-80.
A. Mohebi; H. Mohebi. "Some relations between ε-directional derivative and ε-generalized weak subdifferential". Wavelet and Linear Algebra, 2, 1, 2015, 65-80.
Mohebi, A., Mohebi, H. (2015). 'Some relations between ε-directional derivative and ε-generalized weak subdifferential', Wavelet and Linear Algebra, 2(1), pp. 65-80.
Mohebi, A., Mohebi, H. Some relations between ε-directional derivative and ε-generalized weak subdifferential. Wavelet and Linear Algebra, 2015; 2(1): 65-80.
Some relations between ε-directional derivative and ε-generalized weak subdifferential
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∗⟩, ∀ x∗ ∈ ∂ f(x0) is satisfied. We prove that a similar result can be obtained for ε- generalized weak subdifferential.
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