Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems

In this paper, we consider the problem of minimizing a convex objective which is the sum of three parts: a smooth part, a simple non-smooth Lipschitz part, and a simple non-smooth non-Lipschitz part. A novel optimization algorithm is proposed for solving this problem. By making use of the Gaussian smoothing function of the functions occurring in the objective, we smooth the second part to a convex and differentiable function with Lipschitz continuous gradient by using both variable and constant smoothing parameters. The resulting problem is solved via an accelerated proximal-gradient method and this allows us to recover approximately the optimal solutions to the initial optimization problem with a rate of convergence of order $O(\frac{\ln k}{k})$ for variable smoothing and of order $O(\frac{1}{k})$ for constant smoothing.