  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.
