Splitting Forward-Backward Penalty Scheme for Constrained Variational Problems

We study a forward backward splitting algorithm that solves the variational inequality $$A x +\nabla \Phi(x)+ N_C (x) \ni 0$$ where $\mathcal{H}$ is a real Hilbert space, $A: \mathcal{H}\rightrightarrows \mathcal{H}$ is a maximal monotone operator, $\Phi: \mathcal{H}\to\mathbf{R}$ is a smooth convex function, and $N_C$ is the outward normal cone to a closed convex set $C\subset\mathcal{H}$. The constraint set $C$ is represented as the intersection of the sets of minima of two convex penalization function $\Psi_1:\mathcal{H}\to\mathbf{R}$ and $\Psi_2:\mathcal{H}\to\mathbf{R}\cup \{+\infty\}$. 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), $n\geq 1$, $$\ \left\{\begin{array}{rcl} x_1 & \in & \mathcal{H},\\ x_{n+1} & = & (I+\lambda_n A+\lambda_n\beta_n\partial\Psi_2)^{-1} (x_n-\lambda_n\nabla\Phi(x_n)-\lambda_n\beta_n\nabla\Psi_1(x_n)), \end{array}\right.$$ 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.