Sublinear Convergence Rates of Extragradient-Type Methods: A Survey on Classical and Recent Developments

The generalized equation (also called the [non]linear inclusion) provides a unified template to model various problems in computational mathematics and related fields su ch as the optimality condition of optimization problems (in both unconstrained and constrained settings), minimax optimization, variational inequality, complementarity, two-person game, and fixed-point problem s, see, e.g., [11, 24, 50, 112, 116, 118, 120]. Theory and numerical methods for this equation and its special case s have been extensively studied for many decades, see, e.g., the following monographs and the references quot ed therein [11, 50, 94, 119]. At the same time, several applications of this mathematical tool in operatio ns research, economics, uncertainty quantification, and transportations have been investigated [14, 52, 61, 50, 72]. In the last few years, there has been a surge of research in minimax problems due to new applications in mach ine learning and robust optimization, especially in generative adversarial networks (GANs), adversarial tr aining, and distributionally robust optimization, see, e.g., [4, 14, 55, 76, 84, 114] as a few examples. Minimax probl ems have also found new applications in online learning and reinforcement learning, among many others, se e, e.g., [4, 9, 15, 55, 67, 76, 78, 84, 114, 139]. Such prominent applications have motivated the research in minimax optimization and variational inequality problems (VIPs). On the one hand, classical algorithms such as gradient descent-ascent, extragradient, and primal-dual methods have been revisited, improved, and ext ended. On the other hand, new variants such as accelerated extragradient and accelerated operator split ting schemes have also been developed and equipped with rigorous convergence guarantees and practical perfor mance evaluation. This new development motivates us to write this survey paper, with the focus on sublinear con vergence rate analysis.