A strongly convergent inertial inexact proximal-point algorithm for monotone inclusions with applications to variational inequalities

We propose an inertial variant of the strongly convergent inexact proximal-point (PP) method of Solodov and Svaiter [Forcing strong convergence of proximal point iterations in a Hilbert space. Math Program. 2000;87(1, Ser. A):189–202.] for monotone inclusions. We prove strong convergence of our main algorithm under less restrictive assumptions on the inertial parameters when compared to previous analysis of inertial PP-type algorithms, which makes our method of interest even in finite-dimensional settings. We also performed an iteration-complexity analysis and applied our main algorithm to variational inequalities for monotone operators, obtaining strongly convergent (inertial) variants of Korpolevich's extragradient, forward-backward and Tseng's modified forward-backward methods. Preliminary numerical experiments indicate that our strongly convergent variant of Tseng's modified forward-backward method performs well on certain matrix game problems.