Parallel and Distributed Methods for Constrained Nonconvex Optimization-Part II: Applications in Communications and Machine Learning

In Part I of this paper, we proposed and analyzed a novel algorithmic framework for the minimization of a nonconvex objective function, subject to nonconvex constraints, based on inner convex approximations. This Part II is devoted to the (nontrivial) application of the framework to the following relevant large-scale problems ranging from communications to machine learning: 1) (generalizations of) the rate profile maximization in MIMO interference broadcast networks; 2) the max–min fair multicast multigroup beamforming problem in a multicell environment; and 3) a general nonconvex constrained bi-criteria formulation for k -sparse variable selection in statistical learning; the two criteria are a nonconvex loss objective function, measuring the fitness of the model to data, and the latter is a nonconvex sparsity-inducing constraint in the general form of difference-of-convex (DC) functions, which allows to accomodate in a unified fashion convex and nonconvex surrogates of the ℓ 0 function. The proposed algorithms outperform current state-of-the-art schemes for 1)–3) both theoretically and numerically. For instance, they are the first distributed schemes for the class of problems 1) and 2); and they also lead to subproblems enjoying closed form solutions.