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.

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

1 Introduction- 2 Algorithmic Operators- 3 Convergence of Iterative Methods- 4 Algorithmic Projection Operators- 5 Projection methods

Iterative Methods for Fixed Point Problems in Hilbert Spaces

The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made dynamical systems of coupled oscillators. It is well known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength in the finite- and infinite-dimensional cases and for both first-order and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling strength to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of...

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On the Critical Coupling for Kuramoto Oscillators

The split feasibility problem (SFP) consists in finding a point in a given closed convex subset of a Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another Hilbert space. Iterative methods can be employed to solve the SFP. The most popular iterative method is Byrne’s CQ algorithm. However, to employ Byrne’s CQ algorithm, one needs to know a priori the norm (or at least an estimate of the norm) of the bounded linear operator (matrix in the finite-dimensional framework). It is the purpose of this paper to introduce a way of selecting the stepsizes such that the implementation of the CQ algorithm does not need any prior information about the operator norm. We also practise this way of selecting stepsizes for variants of the CQ algorithm, including a relaxed CQ algorithm where the two closed convex sets are both level sets of convex functions, and a Halpern-type algorithm. Both weak and strong convergence are investigated. Numerical experiments are included to illustrate the applications in signal processing of the CQ algorithm with stepsizes selected in an adaptive way.

https://iopscience.iop.org/article/10.1088/0266-5611/28/8/085004/pdf

Solving the split feasibility problem without prior knowledge of matrix norms

https://hal.univ-antilles.fr/hal-00780778/document

Convergence of a splitting inertial proximal method for monotone operators

This paper deals with an iterative method, in a real Hilbert space, for approximating a common element of the set of fixed points of a demicontractive operator (possibly quasi-nonexpansive or strictly pseudocontractive) and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. The considered algorithm can be regarded as a combination of a variation of the hybrid steepest descent method and the so-called extragradient method. Under classical conditions, we prove the strong convergence of the sequences of iterates given by the considered scheme.

A Hybrid Extragradient-Viscosity Method for Monotone Operators and Fixed Point Problems

Convergence theorems for inertial KM-type algorithms

Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces

In this paper, we discuss the strong convergence of the viscosity approximation method, in Hilbert spaces, relatively to the computation of fixed points of operators in the wide class of quasi-nonexpansive mappings. Our convergence results improve previously known ones obtained for the class of nonexpansive mappings.

Paul-Emile Maingé

Papers

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

A Hybrid Extragradient-Viscosity Method for Monotone Operators and Fixed Point Problems

Convergence theorems for inertial KM-type algorithms

Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces

The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces