Splitting forward-backward penalty scheme for constrained variational problems

We study a forward backward splitting algorithm that solves the variational inequality (equation presented) where H is a real Hilbert space, A : H H is a maximal monotone operator, D : H → R is a smooth convex function, and NC is the outward normal cone to a closed convex set C H. The constraint set C is represented as the intersection of the sets of minima of two convex penalization function ψ1 : H → R and ψ2 : H → R. The function ψ1 is smooth, the function ψ2 is proper and lower semicontinuous. Given a sequence (βn) of penalization parameters which tends to infinity, and a sequence of positive time steps (βn), the algorithm (equation presented) 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 θ 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.