A partially derivative-free cyclic block coordinate descent method for nonseparable composite optimization

In this paper, we address a composite optimization problem in which the objective function consists of two terms: the first presents a function with a gradient that satisfies a Lipschitz–Hölder composition, while the second one is a convex function. Under general settings, we propose and analyze a new coordinate descent method that can operate without the use of derivatives. The algorithm is an adaptation of the coordinate proximal gradient method, specifically designed to consider the composite form of the objective function. We perform a complete worst-case complexity analysis, assuming that the coordinates (or blocks of coordinates) are selected in a cyclic manner. In addition, we present academic numerical examples that illustrate the efficiency of our algorithm in practical problems.