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

Abstract

In this paper, we consider a class of nonconvex and nonsmooth fractional programming problems, that 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. We propose an adaptive full-splitting proximal subgradient algorithm that addresses the challenge of decoupling the composition of the nonsmooth component with the linear operator in the numerator. We specifically evaluate the nonsmooth function in the numerator using its proximal operator of its conjugate function. Furthermore, the smooth component in the numerator is evaluated through its gradient, and the nonsmooth in the denominator is managed using its subgradient. We demonstrate subsequential convergence toward an approximate lifted stationary point and ensure global convergence under the Kurdyka-Łojasiewicz property, all achieved without full-row rank assumptions on the linear operators. We provide further discussions on {\it the tightness of the convergence results of the proposed algorithm and its related variants, and the reasoning behind aiming for an approximate lifted stationary point}. We construct a series of counter-examples to show that the proposed algorithm and its variant might diverge when seeking exact solutions. A practical version incorporating a nonmonotone line search is also developed to enhance its performance significantly. Our theoretical findings are validated through simulations involving limited-angle CT reconstruction and the robust sharp-ratio-type minimization problem.