Robustness of the hybrid extragradient proximal-point algorithm

The hybrid extragradient proximal-point method recently proposed by Solodov and Svaiter has the distinctive feature of allowing a relative error toler ance. We extend the error tolerance of this method, proving that it converg es even if a summable error is added to the relative error. Furthermore, th e extragradient step may be performed inexactly with a summable error. We p resent a convergence analysis, which encompasses other well-known variation s of the proximal-point method, previously unrelated. We establish weak glo bal convergence under mild assumptions.