A hybrid proximal generalized conditional gradient method and application to total variation parameter learning

In this paper we present a new method for solving optimization problems involving the sum of two proper, convex, lower semicontinuous functions, one of which has Lipschitz continuous gradient. The proposed method has a hybrid nature that combines the usual forward-backward and the generalized conditional gradient method. We establish a convergence rate of $o(k^{-1\mathrm{/}3})$ under mild assumptions with a specific step-size rule and show an application to a total variation parameter learning problem, which demonstrates its benefits in the context of nonsmooth convex optimization.