This paper focuses on the study of the second-order directional derivative of a symmetric matrix-valued function of the form F (X) = Pdiag[f(λ1(X)), · · · , f(λn(X))]P . For this purpose, we first adopt a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a matrix in Torki [13]; Second, we establish a formula for the (parabolic) second-order direct...

In this paper, we define the β-partial derivative as well β-directional of a multi-variable function based on β-difference operator, Dβ, which is defined by Dβf(t)=f(β(t))−f(t)/β(t)−t, where β strictly increasing continuous function. Some properties are proved. Furthermore, β-gradient vector and directional introduced. Finally, deduce Hahn-partial Hahn-directional derivatives associated with Ha...

In this paper, we study ε-generalized weak s...

To find a stationary point (i.e., a local maximum or minimum) the derivative ḟ(x) needs to vanish. In higher dimensions, the gradient ∇f needs to be identical to zero. The latter may also be expressed as fu = 0 ∀u. In a sense these simple derivatives are the most basic form of the calculus of variations. Usually, however, one talks about calculus of variations in the context of determining func...

The present paper provides first and second-order characterizations of a radially lower semicontinuous strictly pseudoconvex function f : X → R defined on a convex setX in the real Euclidean space Rn in terms of the lower Dini-directional derivative. In particular we obtain connections between the strictly pseudoconvex functions, nonlinear programming problem, Stampacchia variational inequality...

Given an oblique reflection map Γ and functions ψ, χ ∈ Dlim (the space of R -valued functions that have finite left and right limits at every point), the directional derivative ∇χΓ(ψ) of Γ along χ, evaluated at ψ, is defined to be the pointwise limit, as ε ↓ 0, of the family of functions ∇εχΓ(ψ) . = ε [Γ(ψ + εχ) − Γ(ψ)]. Directional derivatives are shown to exist and lie in Dlim for oblique ref...

In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. We focus on a...