An Equivalent Reformulation and Multiproximity Gradient Algorithms for a Class of Nonsmooth Fractional Programming

We consider a class of structured fractional programs, where the numerator is the sum of a block-separable (possibly nonsmooth nonconvex) function and a locally Lipschitz differentiable (possibly nonconvex) function, and the denominator is a convex (possibly nonsmooth) function. We first present a novel reformulation for the original problem and show the relationship of their optimal solutions, critical points, and Kurdyka-Łojasiewicz (KL) exponents. Inspired by the reformulation, we propose a framework of multiproximity gradient algorithms (MPGA), and show the subsequential convergence analysis for two specific algorithms, namely, cyclic MPGA and randomized MPGA. Moreover, we establish the sequential convergence analysis for cyclic MPGA with the monotone line search (CMPGA_ML) under the KL property. We prove that the corresponding KL exponents are 1/2 for several special cases of the fractional programs, and so, CMPGA_ML exhibits a linear convergence rate. Some preliminary numerical experiment results demonstrate the efficiency of our proposed algorithms. Funding: The work of N. Zhang was supported in part by the National Natural Science Foundation of China [Grant 12271181], by the Guangzhou Basic Research Program [Grant 202201010426], and by the Basic and Applied Basic Research Foundation of Guangdong Province [Grant 2023A1515030046], Department of Science and Technology of Guangdong Province. The work of Q. Li was supported in part by the National Natural Science Foundation of China [Grants 12471098 and 11971499] and the Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University [Grant 2020B1212060032], Department of Science and Technology of Guangdong Province.