  Several new accelerated methods in minimax optimization and fixed-point
iterations have recently been discovered, and, interestingly, they rely on a
mechanism distinct from Nesterov's momentum-based acceleration. In this work,
we show that these accelerated algorithms exhibit what we call the merging path
(MP) property; the trajectories of these algorithms merge quickly. Using this
novel MP property, we establish point convergence of existing accelerated
minimax algorithms and derive new state-of-the-art algorithms for the
strongly-convex-strongly-concave setup and for the prox-grad setup.
