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

Abstract

We are concerned with the study of a class of forward-backward penalty schemes for solving variational inequalities $0\in Ax + N_C (x)$ where $\mathcal{H}$ is a real Hilbert space, $A: \mathcal{H}\rightrightarrows \mathcal{H}$ is a maximal monotone operator, and $N_C$ is the outward normal cone to a closed convex set $C\subset\mathcal{H}$. Let $\Psi: \mathcal{H} \to \mathbb R$ be a convex differentiable function whose gradient is Lipschitz continuous and which acts as a penalization function with respect to the constraint $x\in C.$ Given a sequence $(\beta_n)$ of penalization parameters which tends to infinity, and a sequence of positive time steps $(\lambda_n) \in\ell^2\setminus\ell^1$, we consider the diagonal forward-backward algorithm $x_{n+1}=(I+\lambda_nA)^{-1}(x_n-\lambda_n\beta_n \nabla \Psi (x_n)).$ Assuming that $(\beta_n)$ satisfies the growth condition $\limsup_{n\to\infty}\lambda_n\beta_n<2/\theta$ (where $\theta$ is the Lipschitz constant of $\nabla \Psi$), we obtain weak ergodic convergence of the sequence $(x_n)$ to an equilibrium for a general maximal monotone operator $A$. We also obtain weak convergence of the whole sequence $(x_n)$ when $A$ is the subdifferential of a proper lower-semicontinuous convex function. As a key ingredient of our analysis, we use the cocoerciveness of the operator $\nabla \Psi$. When specializing our results to coupled systems, we bring new light to Passty's theorem and obtain convergence results of new parallel splitting algorithms for variational inequalities involving coupling in the constraint. We also establish robustness and stability results that account for numerical approximation errors. An illustration of compressive sensing is given.