On the Complexity of the Hybrid Proximal Extragradient Method for the Iterates and the Ergodic Mean

In this paper we analyze the iteration-complexity of the hybrid proximal extragradient (HPE) method for finding a zero of a maximal monotone operator recently proposed by Solodov and Svaiter. One of the key points of our analysis is the use of a new termination criteria based on the εenlargement of a maximal monotone operator. The advantages of using this termination criterion it that its definition does not depend on the boundedness of the domain of the operator. We then show that Korpelevich’s extragradient method for solving monotone variational inequalities falls in the framework of the HPE method. As a consequence, using the complexity analysis of the HPE method, we obtain new complexity bounds for Korpelevich’s extragradient method which do not require the feasible set to be bounded, as assumed in a recent paper by Nemirovski. Another feature of our analysis it that the derived iteration complexity bounds are proportional to the distance of the initial point to the solution set. Using also the framework of the HPE method, we study the complexity of a variant of a Newton-type extragradient algorithm proposed by Solodov and Svaiter for finding a zero of a smooth monotone function with Lipschitz continuous Jacobian.