Résumé
This paper is concerned with the study of a class of prox-penalization methods for solving variational inequalities of the form Ax + NC(x) 3 0 where H is a real Hilbert space, A : H ¶ H is a maximal monotone operator and NC is the outward normal cone to a closed convex set C ½ H. Given ª : H ! R [ f+1g which acts as a penalization function with respect to the constraint x 2 C; and a penalization parameter ¯n, we consider a diagonal proximal algorithm of the form xn = ³ I + ¸n(A + ¯n@ª) '¡1 xn¡1; and an algorithm which alternates proximal steps with respect to A and penalization steps with respect to C and reads as xn = (I + ¸n¯n@ª)¡1(I + ¸nA)¡1xn¡1: We obtain weak ergodic convergence for a general maximal monotone operator A, and weak convergence of the whole sequence fxng when A is the subdi®erential of a proper lower- semicontinuous convex function. Mixing with Passty's idea, we can extend the ergodic con- vergence theorem, so obtaining the convergence of a prox-penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone opera- tors. 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.