  In this work we aim to solve a convex-concave saddle point problem, where the
convex-concave coupling function is smooth in one variable and nonsmooth in the
other and not assumed to be linear in either. The problem is augmented by a
nonsmooth regulariser in the smooth component. We propose and investigate a
novel algorithm under the name of OGAProx, consisting of an optimistic gradient
ascent step in the smooth variable coupled with a proximal step of the
regulariser, and which is alternated with a {proximal step} in the nonsmooth
component of the coupling function. We consider the situations convex-concave,
convex-strongly concave and strongly convex-strongly concave related to the
saddle point problem under investigation. Regarding iterates we obtain (weak)
convergence, a convergence rate of order $ \mathcal{O}(\frac{1}{K}) $ and
linear convergence like $\mathcal{O}(\theta^{K})$ with $ \theta < 1 $,
respectively. In terms of function values we obtain ergodic convergence rates
of order $ \mathcal{O}(\frac{1}{K}) $, $ \mathcal{O}(\frac{1}{K^{2}}) $ and $
\mathcal{O}(\theta^{K}) $ with $ \theta < 1 $, respectively. We validate our
theoretical considerations on a nonsmooth-linear saddle point problem, the
training of multi kernel support vector machines and a classification problem
incorporating minimax group fairness.
