  Last-iterate behaviors of learning algorithms in repeated two-player zero-sum
games have been extensively studied due to their wide applications in machine
learning and related tasks. Typical algorithms that exhibit the last-iterate
convergence property include optimistic and extra-gradient methods. However,
most existing results establish these properties under the assumption that the
game is time-independent. Recently, (Feng et al, 2023) studied the last-iterate
behaviors of optimistic and extra-gradient methods in games with a time-varying
payoff matrix, and proved that in an unconstrained periodic game,
extra-gradient method converges to the equilibrium while optimistic method
diverges. This finding challenges the conventional wisdom that these two
methods are expected to behave similarly as they do in time-independent games.
However, compared to unconstrained games, games with constrains are more common
both in practical and theoretical studies. In this paper, we investigate the
last-iterate behaviors of optimistic and extra-gradient methods in the
constrained periodic games, demonstrating that similar separation results for
last-iterate convergence also hold in this setting.
