A full splitting algorithm for fractional programs with structured numerators and denominators

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.