A Parallel Splitting Method for Coupled Monotone Inclusions

Abstract

A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces. Convergence is established for a wide class of coupling schemes. Unlike classical alternating algorithms, which are limited to two variables and linear coupling, our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions, dynamical games, signal recovery, image decomposition, best approximation, network flows, and variational problems in Sobolev spaces.where H 1 , H 2 , and G are Hilbert spaces, f 1 : H 1 → ]−∞, +∞] and f 2 : H 2 → ]−∞, +∞] are proper lower semicontinuous convex functions, and L 1 : H 1 → G and L 2 : H 2 → G are linear and bounded. This problem was solved in [5] via an inertial alternating minimization procedure first proposed in [9] for the case of the strongly coupled problem (1.2).The above problems and their solution algorithms are limited to two variables which, in addition, must be linearly coupled. These are serious restrictions since models featuring more than two variables and/or nonlinear coupling schemes arise naturally in applications. The purpose of this paper is to address simultaneously these restrictions by proposing a parallel algorithm for solving systems of monotone inclusions involving an arbitrary number of variables and nonlinear coupling. The breadth and flexibility of this framework will be illustrated through applications in the areas of evolution inclusions, dynamical games, signal recovery, image decomposition, best approximation, network flows, and decomposition methods in Sobolev spaces.We now state our problem formulation and our standing assumptions.

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

“…The Lipschitz constant of ∇h is denoted by L. This kind of structured problem occurs frequently, see for instance [25,6] and Example 5.4.…”

A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

“…The wealth and applicability of this notion is illustrated through the following examples (one can consult [5] for further examples): Proposition 3.13. Let T : H ⇉ H be a maximal monotone operator.…”

Coupling Forward-Backward with Penalty Schemes and Parallel Splitting for Constrained Variational Inequalities

“…Domains of application. The method developed in this paper can be applied to find Wardrop equilibria for network flows and construct best approximations for the convex feasibility problem (see [5]), as well as domain decomposition methods for PDE's [4] and optimal control problems [7] or best response dynamics for potential games [4]. Here we discuss on the problem of finding the sparsest solutions of underdetermined systems of equations.…”

“…However, they tend to be less stable than the implicit ones. An abundant literature has been devoted to the study of the forward-backward algorithms, and their many applications, see [5,Attouch,Briceño and Combettes], [18,Combettes and Wajs] and the references therein. Thus the main original aspect of our approach is to show how such algorithms can be combined with penalization methods.…”

“…It is a well established fact that the cocoerciveness of the operator, with respect to which the forward step is performed, is a crucial property for obtaining the convergence of the forward-backward algorithm, see [18], [5]. …”