In this paper we present a primal interior-point hybrid proximal extragradient (HPE) method for solving a monotone variational inequality over a closed convex set endowed with a self-concordant barrier and whose underlying map has Lipschitz continuous derivative. In contrast to the method of "R. D. C. Monteiro and B. F. Svaiter, Iteration-Complexity of a Newton Proximal Extrgradient Method for Monotone Variational Inequalities and Inclusion ProblemSIAM J. Optim., 22 (2012), pp. 914-935", in which each iteration required an approximate solution of a linearized variational inequality over the original feasible set, the present one only requires solving a Newton linear system of equations. The method performs two types of iterations, namely: those which follow an ever changing path within a certain ``proximal interior central surface' and those which correspond to a large-step HPE iteration of the type described in the paper above. Due to its first-order nature, the iteration-complexity of the method is shown to be faster than its 0-th order counterparts such as Korpelevich's algorithm and Tseng's modified forward-backward splitting method.
Citation
Research report, Instituto de Matemática, Universidade Federal da Bahia, Salvador, Bahia, Brazil, June 2013.