COUPLING FORWARD-BACKWARD WITH PENALTY SCHEMES AND PARALLEL SPLITTING FOR CONSTRAINED VARIATIONAL INEQUALITIES

We are concerned with the study of a class of forward-backward penalty schemes for solving variational inequalities 0 is an element of Ax + N-C(x) where H is a real Hilbert space, A : H paired right arrows H is a maximal monotone operator, and N-C is the outward normal cone to a closed convex set C subset of H. Let Psi : H -> R be a convex differentiable function whose gradient is Lipschitz continuous and which acts as a penalization function with respect to the constraint x is an element of C. Given a sequence (beta(n)) of penalization parameters which tends to infinity, and a sequence of positive time steps (lambda(n)) is an element of l(2)\l(1), we consider the diagonal forward-backward algorithm x(n+1) = (I + lambda(n)A)(-1)(x(n) - lambda(n)beta(n)del Psi(x(n))). Assuming that (beta(n)) satisfies the growth condition limsup(n ->infinity)lambda(n)beta(n) < 2/theta (where theta is the Lipschitz constant of del 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 del 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.