  In this paper, we consider a class of nonconvex and nonsmooth fractional
programming problems, which involve the sum of a convex, possibly nonsmooth
function composed with a linear operator and a differentiable, possibly
nonconvex function in the numerator and a convex, possibly nonsmooth function
composed with a linear operator in the denominator. These problems have
applications in various fields, including CT reconstruction and sparse signal
recovery. We propose an adaptive full-splitting proximal subgradient algorithm
with an extrapolated step that addresses the challenge of evaluating the
composition in the numerator by decoupling the linear operator from the
nonsmooth component. We specifically evaluate the nonsmooth function using its
proximal operator, while the linear operator is assessed through forward
evaluations. Furthermore, the smooth component in the numerator is evaluated
through its gradient, the nonsmooth component in the denominator is managed
using its subgradient, and the linear operator in the denominator is also
assessed through forward evaluations. We demonstrate subsequential convergence
toward an approximate lifted stationary point and ensure global convergence
under the Kurdyka-\L ojasiewicz property, all achieved {\it without relying on
any full-row rank assumptions regarding the linear operators}. We further
explain the reasoning behind aiming for an approximate lifted stationary point.
This is exemplified by constructing a scenario illustrating that the algorithm
could diverge when seeking exact solutions. Lastly, we present a practical
iteration of the algorithm incorporating a nonmonotone line search,
significantly enhancing its convergence performance. Our theoretical findings
are validated through simulations involving limited-angle CT reconstruction and
the robust sharp ratio minimization problem.
