Splitting forward-backward penalty scheme for constrained variational problems

We study a forward backward splitting algorithm that solves the variational inequality beginequation* A x +nabla Phi(x)+ NC (x) ni 0 endequation* where H is a real Hilbert space, A: Hrightrightarrows H is a maximal monotone operator, Phi: H→ℝ is a smooth convex function, and NC is the outward normal cone to a closed convex set Csubset H. The constraint set C is represented as the intersection of the sets of minima of two convex penalization function Psi1:H→ℝ and Psi2: H→ℝcup +∞. The function Psi1 is smooth, the function Psi2 is proper and lower semicontinuous. Given a sequence (βn) of penalization parameters which tends to infinity, and a sequence of positive time steps (λn), the algorithm leftbeginarrayrcl x1 & ∈ & H, xn+1 & = & (I+λn A+λnβnpartialPsi2)-1(xn-λnnablaPhi(xn)-λnβnnablaPsi1(xn)), n≥ 1. endarrayright. 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 (xn) 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 (xn) 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.