Accelerating block-decomposition first-order methods for solving generalized saddle-point and Nash equilibrium problems

This article considers the generalized (two-player) Nash equilibrium (GNE) problem with a separable non-smooth part, which is known to include the generalized saddle-point (GSP) problem as a special case. Due to its two-block structure, this problem can be solved by any algorithm belonging to the block-decomposition hybrid proximal-extragradient framework proposed in [13]. The framework consists of a family of inexact proximal point methods based on relative error criteria for solving monotone inclusion problems with a two-block structure. This paper focuses in particular on two algorithms from this framework. By choosing the proximal stepsize constant and sufficiently small, the first method approximately solves the prox subinclusion by performing a single resolvent evaluation. This method resembles, and in fact has similar iteration-complexity as, Tseng’s modified forward-backward splitting algorithm [26]. The second method exploits the fact that the two prox sub-inclusions are equivalent to composite convex programs and uses a Nesterov-type accelerated method (e.g., [17]) to approximately solve them. As a result, it is able to take a (still constant but) significantly larger stepsize than the first method. Finally, it is shown that the second method substantially outperforms the first one both theoretically and computationally on many relevant GSP and GNE instances.