Prox-Penalization and Splitting Methods for Constrained Variational Problems

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 ⊂ H. Given Ψ : H → R ∪ {+∞} which acts as a penalization function with respect to the constraint x ∈ C, and a penalization parameter βn, we consider a diagonal proximal algorithm of the form xn = ( I + λn(A + βn∂Ψ) )−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 + λnβn∂Ψ) −1(I + λnA) xn−1. We obtain weak ergodic convergence for a general maximal monotone operator A, and weak convergence of the whole sequence {xn} when A is the subdifferential of a proper lowersemicontinuous convex function. Mixing with Passty’s idea, we can extend the ergodic convergence theorem, so 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. Introduction Let H be a real Hilbert space, A : H ⇒ H a general maximal monotone operator, and C a closed convex set in H. We denote by NC the outward normal cone to C. This paper is concerned with the study of a class of prox-penalization and splitting algorithms for solving variational inequalities of the form (1) Ax + NC(x) 3 0, which combine proximal steps with respect to A and penalization steps with respect to C. We begin by describing two model situations that motivate our study: 1. Sum of maximal monotone operators. Let X be a real Hilbert space and setH = X×X . Define A : H ⇒ H by A(x1, x2) = (A1x1, A2x2) where A1 and A2 are maximal monotone operators on X . If C = {(x1, x2) ∈ X × X : x1 = x2} the inclusion (1) reduces to (2) A1x + A2x 3 0. Date: March, 15, 2010. 1991 Mathematics Subject Classification. 37N40, 46N10, 49M30, 65K05, 65K10,90B50, 90C25.