  In this paper, we consider a class of structured nonconvex nonsmooth
optimization problems, in which the objective function is formed by the sum of
a possibly nonsmooth nonconvex function and a differentiable function whose
gradient is Lipschitz continuous, subtracted by a weakly convex function. This
type of structured problems has many practical applications in machine learning
and statistics such as compressed sensing, signal recovery, sparse dictionary
learning, clustering, matrix factorization, and others. We develop a flexible
extrapolated proximal subgradient algorithm for solving these problems with
guaranteed subsequential convergence to a stationary point. The global
convergence of the whole sequence generated by our algorithm is also
established under the Kurdyka-Lojasiewicz property. To illustrate the promising
numerical performance of the proposed algorithm, we conduct numerical
experiments on two important nonconvex models. This includes a least squares
problem with a nonconvex regularization and an optimal power flow problem with
distributed energy resources.
