  In this paper we first present a novel operator extrapolation (OE) method for
solving deterministic variational inequality (VI) problems. Similar to the
gradient (operator) projection method, OE updates one single search sequence by
solving a single projection subproblem in each iteration. We show that OE can
achieve the optimal rate of convergence for solving a variety of VI problems in
a much simpler way than existing approaches. We then introduce the stochastic
operator extrapolation (SOE) method and establish its optimal convergence
behavior for solving different stochastic VI problems. In particular, SOE
achieves the optimal complexity for solving a fundamental problem, i.e.,
stochastic smooth and strongly monotone VI, for the first time in the
literature. We also present a stochastic block operator extrapolations (SBOE)
method to further reduce the iteration cost for the OE method applied to
large-scale deterministic VIs with a certain block structure. Numerical
experiments have been conducted to demonstrate the potential advantages of the
proposed algorithms. In fact, all these algorithms are applied to solve
generalized monotone variational inequality (GMVI) problems whose operator is
not necessarily monotone. We will also discuss optimal OE-based policy
evaluation methods for reinforcement learning in a companion paper.
