  The monotone variational inequality is a central problem in mathematical
programming that unifies and generalizes many important settings such as smooth
convex optimization, two-player zero-sum games, convex-concave saddle point
problems, etc. The extragradient algorithm by Korpelevich [1976] and the
optimistic gradient descent-ascent algorithm by Popov [1980] are arguably the
two most classical and popular methods for solving monotone variational
inequalities. Despite their long histories, the following major problem remains
open. What is the last-iterate convergence rate of the extragradient algorithm
or the optimistic gradient descent-ascent algorithm for monotone and Lipschitz
variational inequalities with constraints? We resolve this open problem by
showing that both the extragradient algorithm and the optimistic gradient
descent-ascent algorithm have a tight $O\left(\frac{1}{\sqrt{T}}\right)$
last-iterate convergence rate for arbitrary convex feasible sets, which matches
the lower bound by Golowich et al. [2020a,b]. Our rate is measured in terms of
the standard gap function. At the core of our results lies a non-standard
performance measure -- the tangent residual, which can be viewed as an
adaptation of the norm of the operator that takes the local constraints into
account. We use the tangent residual (or a slight variation of the tangent
residual) as the the potential function in our analysis of the extragradient
algorithm (or the optimistic gradient descent-ascent algorithm) and prove that
it is non-increasing between two consecutive iterates.
