Search

In a two-player noncooperative game framework, we deal with the affine relaxations of the best response algorithm (where a player's strategy is a best response to the strategy of the other player that comes from the previous step), motivated by the first results obtained for convex relaxations in zero-sum games (J. Morgan, Int. J. Comput. Math., 4 (1974), pp. 143-175) and for nonconvex affine relaxations in non-zero-sum games (F. Caruso, M. C. Ceparano, and J. Morgan, SIAM J. Optim., 30 (2020), pp. 1638-1663). In order to be able to specify the convergence of any type of affine relaxation of the best response algorithm, we define a new class of games, called ratiobounded games. This class contains games broadly used in literature (such as weighted potential and zero-sum games), both in finite and infinite dimensional settings. Its definition relies on a unifying property and on three associate key parameters explicitly related to the data. Depending on how the parameters are ordered, we provide a classification of the ratio-bounded games in four subclasses such that, for each of them, the following issues are answered when the strategy sets are real Hilbert spaces: existence and uniqueness of the Nash equilibria, global convergence of the affine relaxations of the best response algorithm, estimation of the related errors, determination of the algorithm with the highest speed of convergence, and comparison with the known results. Moreover, the investigation is supplemented by illustrating numerical examples and by describing a black-box model with a first-order oracle for an implementation of the algorithm. [ABSTRACT FROM AUTHOR]

We provide improved convergence rates for constrained convex-concave min-max problems and monotone variational inequalities with higher-order smoothness. In min-max settings where the pth-order derivatives are Lipschitz continuous, we give an algorithm that achieves an iteration complexity of O(1/T p+2/2) when given access to an oracle for finding a fixed point of a pth-order equation. We give analogous rates for the weak monotone variational inequality problem. For p > 2, our results improve upon the iteration complexity of the first-order Mirror Prox method by Nemirovski [SIAM J. Optim., 15 (2004), pp. 229-251] and the second-order method by Monteiro and Svaiter [SIAM J. Optim., 22 (2012), pp. 914-935]. We further instantiate our entire algorithm in the unconstrained p = 2 case. [ABSTRACT FROM AUTHOR]