针对分式l₁正则化优化问题的自适应ADMM算法

Adaptive ADMM for Optimization Problems with Fractional l₁ Regularization

摘要: 本文研究了一种基于分式$l_1$ 正则化的非凸优化模型，该模型能比传统$l_1$ 范数更精确地逼近$l_0$ 范数的稀疏性。为高效求解此非凸优化问题，我们设计了基于交替方向乘子法的求解框架，并提出了一种自适应的惩罚参数与正则化参数调整策略。该策略能动态调整惩罚参数以平衡原始残差与对偶残差，加速收敛；同时，能自适应地调整正则化参数以维持数据拟合项与正则项的平衡，有效引导迭代过程趋向高质量解。数值实验表明，与现有快速算法相比，本文的方法在求解精度上具有明显的优势，能够稳定地获得更低相对误差的解。而且，该自适应策略增强了对超参数初始设置的鲁棒性，使算法在宽松的参数条件下仍能保持较好的恢复性能，为解决分式$l_1$ 正则化优化问题提供了一个高精度且实用的解决方案。

Abstract: This paper investigates a nonconvex optimization model based on fractional $l_1$ regularization, which provides a superior approximation to the sparsity of the $l_0$ norm compared to conventional $l_1$ norm. To address this, our solution is built around the Alternating Direction Method of Multipliers (ADMM) framework, augmented with a novel strategy for the adaptive tuning of both penalty and regularization parameters. This strategy dynamically balances the primal and dual residuals to accelerate convergence, while adaptively maintaining the equilibrium between the data fidelity and regularization terms, thereby steering the iterations toward high-quality solutions. Our numerical tests show that the proposed method significantly outperforms existing fast algorithms in solution accuracy, reliably achieving lower relative errors. It also exhibits greater robustness, as the adaptive scheme reduces sensitivity to initial hyperparameter choices, allowing it to perform well even with less stringent parameter tuning. Consequently, our approach provides a highly practical and precise solver for problems involving fractional regularization.