带有绝对值形式的非线性约束下的交替方向乘子法
Alternating Direction Method of Multipliers under Nonlinear Constraints with Absolute Value Form
摘要: 交替方向乘子法(Alternating Direction Method of Multipliers, ADMM)广泛应用于线性约束优化问题，无论是凸目标函数还是非凸目标函数，其约束一般是线性约束。文章研究了具有绝对值形式的非线性约束的非凸极小化问题的ADMM。在假设相关函数满足Kurdyka-Lojasiewicz (KL)不等式的情况下，证明了由ADMM生成的迭代子序列收敛于问题的一个临界点，并根据一些数值例子说明了ADMM是可行的。
Abstract: The Alternating Direction Method of Multipliers (ADMM) is widely used for linear constraint optimization problems, whether the objective function is convex or non-convex, with constraints generally being linear. This article studies the ADMM applied to non-convex minimization problems with nonlinear constraints in the form of absolute values. Under the assumption that the relevant functions satisfy the Kurdyka-Lojasiewicz (KL) inequality, it is proved that the iterative subsequence generated by ADMM converges to a critical point of the problem. Additionally, some numerical examples are provided to illustrate the feasibility of ADMM.
文章引用：叶景然, 陈跨越, 杨硕, 苏映梅. 带有绝对值形式的非线性约束下的交替方向乘子法[J]. 应用数学进展, 2025, 14(3): 382-397. https://doi.org/10.12677/aam.2025.143126

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