求解非凸非光滑优化问题的一类惯性Moreau包络交替方向乘子法
An Inertial Moreau Envelope Alternating Direction Method of Multipliers for Solving Nonconvex Nonsmooth Optimization Problems
摘要: 针对一类带有线性约束的非凸非光滑优化问题，本文提出了一类惯性Moreau包络交替方向乘子法(IMADMM)。该方法通过引入Moreau包络对非光滑项进行光滑近似，并嵌入惯性步以加速迭代；另外，在每次迭代中采用Armijo线搜索确定步长，在保持算法稳定性的同时提升了收敛速度。在理论分析方面，在Kurdyka-Łojasiewicz (KŁ)框架下证明了算法的全局收敛性。最后，数值实验以稀疏最小二乘问题为例，验证了所提算法的有效性和优越性。
Abstract: This paper proposes an inertial Moreau envelope alternating direction method of multipliers (IMADMM) for a class of nonconvex nonsmooth optimization problems with linear constraints. The method introduces the Moreau envelope to smoothly approximate the nonsmooth term and embeds an inertial step to accelerate the iteration. Moreover, an Armijo line search is employed at each iteration to determine the step size, enhancing the convergence rate while maintaining algorithmic stability. Theoretically, the global convergence of the algorithm is established under the Kurdyka-Łojasiewicz (KŁ) framework. Finally, numerical experiments on sparse least squares problems validate the effectiveness and superiority of the proposed method.
文章引用：史梦蝶. 求解非凸非光滑优化问题的一类惯性Moreau包络交替方向乘子法[J]. 应用数学进展, 2026, 15(3): 237-252. https://doi.org/10.12677/aam.2026.153102

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