  In this paper, we provide a generalization of the forward-backward splitting
algorithm for minimizing the sum of a proper convex lower semicontinuous
function and a differentiable convex function whose gradient satisfies a
locally Lipschitztype condition. We prove the convergence of our method and
derive a linear convergence rate when the differentiable function is locally
strongly convex. We recover classical results in the case when the gradient of
the differentiable function is globally Lipschitz continuous and an already
known linear convergence rate when the function is globally strongly convex. We
apply the algorithm to approximate equilibria of variational mean field game
systems with local couplings. Compared with some benchmark algorithms to solve
these problems, our numerical tests show similar performances in terms of the
number of iterations but an important gain in the required computational time.
