发表机构
China University of Mining and Technology; RMIT University; Guangxi Minzu University（中国矿业大学; 皇家墨尔本理工大学; 广西民族大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带非线性耦合约束的非凸分式规划问题，提出Bregman邻近线性化ADMM，证明其收敛性与收敛速率，数值实验验证了算法有效性。
AI 中文摘要
本文研究一类具有复合结构和非线性等式约束的非凸分式优化问题，将其重新构造为非分式极小极大问题并得到对应最优解。通过线性化增广拉格朗日中的非线性约束项，在交替方向乘子法（ADMM）中引入Bregman邻近线性化和取值于(0,2)的松弛因子，提出一种Bregman邻近线性化ADMM，可保证其序列收敛至提升型临界点。在Kurdyka-Łojasiewicz（KL）性质下建立了全序列的收敛性，进一步在Hölder型值邻近误差界条件或KL性质下推导了收敛速率，对应指数取值于[0,1)；特别地，当指数在(0,1/2)时建立了超线性收敛，其余情况则对应有限、线性和次线性收敛。数值实验验证了所提算法的有效性。
英文摘要
This paper considers a nonconvex fractional optimization problem with composite structure and nonlinear equality constraints. We reformulate the problem as a non-fractional min-max problem with corresponding optimal solutions. By linearizing the nonlinear constraint terms in the augmented Lagrangian and incorporating Bregman proximal linearization and a relaxation factor in $(0, 2)$ into the alternating direction method of multipliers (ADMM), we develop a Bregman proximal linearized ADMM with guaranteed subsequential convergence to a lifted critical point. Convergence of the entire sequence is established under the Kurdyka--Łojasiewicz (KL) property. We further derive convergence rates under either a Hölderian value proximity error bound condition or the KL property, with the corresponding exponent in $[0, 1)$. In particular, we establish superlinear convergence for exponents in $(0, 1/2)$, together with finite, linear, and sublinear convergence in the remaining cases. Numerical experiments demonstrate the effectiveness of the proposed algorithm.
