  We study first-order methods for constrained min-max optimization. Existing
methods either require two gradient calls or two projections in each iteration,
which may be costly in some applications. In this paper, we first show that a
variant of the Optimistic Gradient (OG) method, a single-call single-projection
algorithm, has $O(\frac{1}{\sqrt{T}})$ best-iterate convergence rate for
inclusion problems with operators that satisfy the weak Minty variation
inequality (MVI). Our second result is the first single-call single-projection
algorithm -- the Accelerated Reflected Gradient (ARG) method that achieves the
optimal $O(\frac{1}{T})$ last-iterate convergence rate for inclusion problems
that satisfy negative comonotonicity. Both the weak MVI and negative
comonotonicity are well-studied assumptions and capture a rich set of
non-convex non-concave min-max optimization problems. Finally, we show that the
Reflected Gradient (RG) method, another single-call single-projection
algorithm, has $O(\frac{1}{\sqrt{T}})$ last-iterate convergence rate for
constrained convex-concave min-max optimization, answering an open problem of
[Heish et al, 2019]. Our convergence rates hold for standard measures such as
the tangent residual and the natural residual.
