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

• 成果类型: Article; Early Access
• 署名作者: Zhou, Junpeng; Zhang, Na; Li, Qia
• 署名单位: Sun Yat Sen University; South China Agricultural University; Sun Yat Sen University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2024.0457
• 发表日期: 2024
• 关键词: error-bounds; proximal algorithm; optimization; minimization; CONVERGENCE; difference; nonconvex

摘要

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 & Lstrok;ojasiewicz (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.