Prox-Penalization and Splitting Methods for Constrained Variational Problems

Abstract

This paper is concerned with the study of a class of prox-penalization methods for solving variational inequalities of the form $Ax+N_C(x)\ni0$, where $\mathcal{H}$ is a real Hilbert space, $A:\mathcal{H}\rightrightarrows\mathcal{H}$ is a maximal monotone operator, and $N_C$ is the outward normal cone to a closed convex set $C\subset\mathcal{H}$. Given $\Psi:\mathcal{H}\to\mathbb{R}\cup\{+\infty\}$, which acts as a penalization function with respect to the constraint $x\in C$, and a penalization parameter $\beta_n$, we consider a diagonal proximal algorithm of the form $x_n=(I+\lambda_n(A+\beta_n\partial\Psi))^{-1}\,x_{n-1}$ and an algorithm which alternates proximal steps with respect to A and penalization steps with respect to C and reads as $x_n=(I+\lambda_n\beta_n\partial\Psi)^{-1}(I+\lambda_nA)^{-1}x_{n-1}$. We obtain weak ergodic convergence for a general maximal monotone operator A and weak convergence of the whole sequence $\{x_n\}$ when A is the subdifferential of a proper lower-semicontinuous convex function. Mixing with Passty's idea, we can extend the ergodic convergence theorem, thus obtaining the convergence of a prox-penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone operators. Our results are applied to an optimal control problem where the state variable and the control are coupled by an elliptic equation. We also establish robustness and stability results that account for numerical approximation errors.