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

• Authors: Hédy Attouch;Marc-Olivier Czarnecki;Juan Peypouquet
• Venue: SIAM Journal on Optimization
• Year: 2011

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