  Variational inequalities provide a framework through which many optimisation
problems can be solved, in particular, saddle-point problems. In this paper, we
study modifications to the so-called Golden RAtio ALgorithm (GRAAL) for
variational inequalities -- a method which uses a fully explicit adaptive
step-size, and provides convergence results under local Lipschitz assumptions
without requiring backtracking. We present and analyse two Bregman
modifications to GRAAL: the first uses a fixed step-size and converges under
global Lipschitz assumptions, and the second uses an adaptive step-size rule.
Numerical performance of the former method is demonstrated on a bimatrix game
arising in network communication, and of the latter on two problems, namely,
power allocation in Gaussian communication channels and $N$-person Cournot
completion games. In all of these applications, an appropriately chosen Bregman
distance simplifies the projection steps computed as part of the algorithm.
