A first-order primal-dual method with adaptivity to local smoothness

We consider the problem of finding a saddle point for the convex-concave objective minx maxy f(x) + langle Ax, yrangle - g^*(y), where f is a convex function with locally Lipschitz gradient and g is convex and possibly non-smooth. We propose an adaptive version of the Condat-V~u algorithm, which alternates between primal gradient steps and dual proximal steps. The method achieves stepsize adaptivity through a simple rule involving |A| and the norm of recently computed gradients of f. Under standard assumptions, we prove an O(k-1) ergodic convergence rate. Furthermore, when f is also locally strongly convex and A has full row rank we show that our method converges with a linear rate. Numerical experiments are provided for illustrating the practical performance of the algorithm.