Set-valued Analysis 

About: Set-valued Analysis is an academic journal. The journal publishes majorly in the area(s): Banach space & Subderivative. It has an ISSN identifier of 0927-6947. Over the lifetime, 384 publications have been published receiving 9739 citations.

Strong Convergence of Projected Subgradient Methods for Nonsmooth and Nonstrictly Convex Minimization

Abstract: In this paper, we establish a strong convergence theorem regarding a regularized variant of the projected subgradient method for nonsmooth, nonstrictly convex minimization in real Hilbert spaces. Only one projection step is needed per iteration and the involved stepsizes are controlled so that the algorithm is of practical interest. To this aim, we develop new techniques of analysis which can be adapted to many other non-Fejerian methods.

An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping

Abstract: The ‘heavy ball with friction’ dynamical system x + γx + ∇f(x)=0 is a nonlinear oscillator with damping (γ>0). It has been recently proved that when H is a real Hilbert space and f: H→R is a differentiable convex function whose minimal value is achieved, then each solution trajectory t→x(t) of this system weakly converges towards a solution of ∇f(x)=0. We prove a similar result in the discrete setting for a general maximal monotone operator A by considering the following iterative method: x k+1−x k −α k (x k −x k−1)+λ k A(x k+1)∋0, giving conditions on the parameters λ k and α k in order to ensure weak convergence toward a solution of 0∈A(x) and extending classical convergence results concerning the standard proximal method.

On the convergence of von Neumann's alternating projection algorithm for two sets

Abstract: We give several unifying results, interpretations, and examples regarding the convergence of the von Neumann alternating projection algorithm for two arbitrary closed convex nonempty subsets of a Hilbert space. Our research is formulated within the framework of Fejer monotonicity, convex and set-valued analysis. We also discuss the case of finitely many sets.

A hybrid approximate extragradient - proximal point algorithm using the enlargement of a maximal monotone operator

Abstract: We propose a modification of the classical extragradient and proximal point algorithms for finding a zero of a maximal monotone operator in a Hilbert space. At each iteration of the method, an approximate extragradient-type step is performed using information obtained from an approximate solution of a proximal point subproblem. The algorithm is of a hybrid type, as it combines steps of the extragradient and proximal methods. Furthermore, the algorithm uses elements in the enlargement (proposed by Burachik, Iusem and Svaiter) of the operator defining the problem. One of the important features of our approach is that it allows significant relaxation of tolerance requirements imposed on the solution of proximal point subproblems. This yields a more practical proximal-algorithm-based framework. Weak global convergence and local linear rate of convergence are established under suitable assumptions. It is further demonstrated that the modified forward-backward splitting algorithm of Tseng falls within the presented general framework.

On Attractors of Multivalued Semi-Flows and Differential Inclusions

Abstract: In this paper we study the existence of global attractors for multivalued dynamical systems. These theorems are then applied to dynamical systems generated by differential inclusions for which the solution is not unique for a given initial state. Finally, some boundary-value problems are considered.