Alternating proximal algorithms for linearly constrained variational inequalities: Application to domain decomposition for PDE’s

Let X,Y,ZX,Y,Z be real Hilbert spaces, let f:X→R∪{+∞}f:X→R∪{+∞}, g:Y→R∪{+∞}g:Y→R∪{+∞} be closed convex functions and let A:X→ZA:X→Z, B:Y→ZB:Y→Z be linear continuous operators. Let us consider the constrained minimization problem (P)min{f(x)+g(y):Ax=By}. Given a sequence (γn)(γn) which tends toward 00 as n→+∞n→+∞, we study the following alternating proximal algorithm (A){xn+1=argmin{γn+1f(ζ)+12‖Aζ−Byn‖Z2+α2‖ζ−xn‖X2;ζ∈X}yn+1=argmin{γn+1g(η)+12‖Axn+1−Bη‖Z2+ν2‖η−yn‖Y2;η∈Y}, where αα and νν are positive parameters. It is shown that if the sequence (γn)(γn) tends moderately slowly   toward 00, then the iterates of (A)(A) weakly converge toward a solution of (P)(P). The study is extended to the setting of maximal monotone operators, for which a general ergodic convergence result is obtained. Applications are given in the area of domain decomposition for PDE’s.