and Applied Analysis 3 the uniform r-prox-regularity C is equivalent to the convexity of C. Moreover, it is clear that the class of uniformly prox-regular sets is sufficiently large to include the class p-convex sets, C1,1 submanifolds possibly with boundary of H, the images under a C1,1 diffeomorphism of convex sets, and many other nonconvex sets; see 6, 8 . Now, let us state the following fac...

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C. Here a corresponding local theory is developed for the property of dC being continuously differentiable outside of C on some neighborhood of a point x ∈ C. This is shown to be equ...

In this paper, we propose a simple variant of the original SVRG, called variance reduced stochastic gradient descent (VR-SGD). Unlike the choices of snapshot and starting points in SVRG and its proximal variant, Prox-SVRG, the two vectors of VR-SGD are set to the average and last iterate of the previous epoch, respectively. The settings allow us to use much larger learning rates, and also make ...

This paper deals with the generalized regularization proximal point method which was introduced by the authors in [Four parameter proximal point algorithms, Nonlinear Anal. 74 (2011), 544-555]. It is shown that sequences generated by it converge strongly under minimal assumptions on the control parameters involved. Thus the main result of this paper unify many results related to the prox-Tikhon...

Active set algorithms, such as the projected gradient method in nonlinear optimization, are designed to “identify” the active constraints of the problem in a finite number of iterations. Using the notions of “partial smoothness” and “prox-regularity” we extend work of Burke, Moré and Wright on identifiable surfaces from the convex case to a general nonsmooth setting. We further show how this se...

We introduce a model-based excessive gap technique to analyze first-order primaldual methods for constrained convex minimization. As a result, we construct new primal-dual methods with optimal convergence rates on the objective residual and the primal feasibility gap of their iterates separately. Through a dual smoothing and prox-function selection strategy, our framework subsumes the augmented...

In this paper, some systems of quasi-variational inequality problems are considered on a class of nonconvex sets, as uniformly prox-regular sets. Some sufficient conditions for the existence solution of the considered problems are provided. Also, some interesting remarks are discussed. The results which are presented in this paper are more general, and may be viewed as an extension, improvement...

OBJECTIVES To evaluate, in patients with ulnar neuropathy at the elbow (UNE), if ultrasonographic differences in ulnar nerve size correlate with severity score determined by electrodiagnostic studies. METHODS We examined prospectively 38 patients (50 elbows) with UNE. Patients were classified into mild, moderate and severe groups according to electrodiagnostic studies. Cross-sectional areas (...