The space of directed sets is a Banach space in which convex compact subsets of Rn are embedded. Each directed set is visualized as a (nonconvex) subset of Rn, which is comprised of a convex, a concave and a mixed-type part. Following an idea of A. Rubinov, the directed subdifferential of a difference of convex (DC) functions is defined as the directed difference of the corresponding embedded c...

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion u′(t)+Au(t)+ι∗∂J(ιu(t)) 3 f(t), where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of...

Abstract Let W (A) denote the field of values (numerical range) of a matrix A. For any polynomial p and matrix A, define the Crouzeix ratio to have numerator max {|p(ζ)| : ζ ∈ W (A)} and denominator ‖p(A)‖2. M. Crouzeix’s 2004 conjecture postulates that the globally minimal value of the Crouzeix ratio is 1/2, over all polynomials p of any degree and matrices A of any order. We derive the subdif...

There exists a calculus for general nondifferentiable functions that englobes a large part of the familiar subdifferential calculus for convex nondifferentiable functions [1]. This development started with F.H. Clarke, who introduced a generalized gradient for functions that are locally Lipschitz, but (possibly) nondifferentiable. Generalized gradients turn out to be the subdifferentials, in th...

Here, 2 ≤ p < ∞, j : Z × R → R is a function which is measurable in z ∈ Z and locally Lipschitz in x ∈ R and ∂ j(z,x) is the Clarke subdifferential of j(z, ·). If f : Z × R → R is a measurable function which is in general discontinuous in the x ∈ R variable, for almost all z ∈ Z, all M > 0, and all |x| ≤ M, we have | f (z,x)| ≤ aM(z) with aM ∈ L1(Z) and we set j(z,x) = ∫x 0 f (z, r)dr, then j(z...

In convex programming, characterizations of the solution set in terms of the subdifferential have been investigated by Mangasarian. An invariance property of the subdifferential of the objective function is studied, and as a consequence, characterizations of the solution set by any solution point and any point in the relative interior of the solution set are given. In quasiconvex programming, h...

We study on a product Banach space the properties of a class of saddle functions called partially ball weakly inf-compact. For such a function we prove that the domain of the subdifferential is nonempty, that the operator naturally associated with the subdifferential is maximal monotone, and that the subdifferential of the function is integrable. For a function in a large subclass of that class...

Some properties of the weak subdifferential are considered in this paper. By using the definition and properties of the weak subdifferential which are described in the papers Azimov and Gasimov, 1999; Kasimbeyli and Mammadov, 2009; Kasimbeyli and Inceoglu, 2010 , the author proves some theorems connecting weak subdifferential in nonsmooth and nonconvex analysis. It is also obtained necessary op...

In this paper, we investigate relation between weak subdifferential and augmented normal cone. We define augmented normal cone via weak subdifferential and vice versa. The necessary conditions for the global maximum are also stated. We produce preliminary properties of augmented normal cones and discuss them via the distance function. Then we obtain the augmented normal cone for the indicator f...