Splitting Forward-Backward Penalty Scheme for Constrained Variational Problems

We study a forward backward splitting algorithm that solves the variational inequality Ax + del phi(x) + N-C(x) (sic) 0 where H is a real Hilbert space, A : H paired right arrows H is a maximal monotone operator, phi : H -> R is a smooth convex function, and N-C is the outward normal cone to a closed convex set C subset of H. The constraint set C is represented as the intersection of the sets of minima of two convex penalization function Psi(1) : H -> R and Psi(2) : H -> R boolean OR {+infinity}. The function Psi(1) is smooth, the function Psi(2) is proper and lower semicontinuous. Given a sequence (beta(n)) of penalization parameters which tends to infinity, and a sequence of positive time steps (lambda(n)), the algorithm (SFBP) {x(1) epsilon H, x(n+1) = (I + lambda(n)A + lambda(n)beta(n)partial derivative Psi(2))(-1) (x(n) - lambda(n)del phi(x(n)) - lambda(n)beta(n)del Psi(1)(x(n))), n >= 1, performs forward steps on the smooth parts and backward steps on the other parts. Under suitable assumptions, we obtain weak ergodic convergence of the sequence (x(n)) to a solution of the variational inequality. Convergence is strong when either A is strongly monotone or phi is strongly convex. We also obtain weak convergence of the whole sequence (x(n)) when A is the subdifferential of a proper lower semicontinuous convex function. This provides a unified setting for several classical and more recent results, in the line of historical research on continuous and discrete gradient-like systems.