Extrapolated Proximal Subgradient Algorithms for Nonconvex and Nonsmooth Fractional Programs

Abstract: In this paper, we consider a broad class of nonsmooth and nonconvex fractional programs, where the numerator can be written as the sum of a continuously differentiable convex function whose gradient is Lipschitz continuous and a proper lower semicontinuous (possibly nonconvex) function, and the denominator is weakly convex over the constraint set. This model problem includes the composite optimization problems studied extensively lately, and encompasses many important modern fractional optimization problems arising from diverse areas such as the recently proposed scale invariant sparse signal reconstruction problem in signal processing. We propose a proximal subgradient algorithm with extrapolations for solving this optimization model and show that the iterated sequence generated by the algorithm is bounded and any of its limit points is a stationary point of the model problem. The choice of our extrapolation parameter is flexible and includes the popular extrapolation parameter adopted in the restarted Fast Iterative Shrinking-Threshold Algorithm (FISTA). By providing a unified analysis framework of descent methods, we establish the convergence of the full sequence under the assumption that a suitable merit function satisfies the Kurdyka--{\L}ojasiewicz (KL) property. In particular, our algorithm exhibits linear convergence for the scale invariant sparse signal reconstruction problem and the Rayleigh quotient problem over spherical constraint. In the case where the denominator is the maximum of finitely many continuously differentiable weakly convex functions, we also propose an enhanced extrapolated proximal subgradient algorithm with guaranteed convergence to a stronger notion of stationary points of the model problem. Finally, we illustrate the proposed methods by both analytical and simulated numerical examples.