具有双重松弛项的改进惯性近端交替方向乘子法在结构化非凸和非光滑问题中的应用

Application of the Modified Inertial Proximal Alternating Direction Method of Multipliers with Dual-Relaxed Term for Structured Nonconvex and Nonsmooth Problems

摘要: 针对结构化的非凸非光滑优化问题，提出了一种改进的惯性近端交替方向乘子法(Modified Inertial Proximal Alternating Direction Method of Multipliers, MID-PADMM)。该问题在多个领域，包括机器学习、信号处理和经济学中具有重要应用。现有算法在处理这类问题时，往往面临收敛速度慢或无法保证收敛的挑战。为了克服这些限制，引入了一种双重松弛项，以增强算法的鲁棒性和灵活性。理论分析表明，MID-PADMM算法在适当的条件下能够实现全局收敛，并且具有O(1/k)的迭代复杂度，其中k代表迭代次数。数值实验结果表明，与现有的状态最优算法相比，MID-PADMM在多个实例中展现出更快的收敛速度和更高的求解质量。

Abstract: For structured nonconvex and nonsmooth optimization problems, we propose a Modified Inertial Proximal Alternating Direction Method of Multipliers (MID-PADMM). These problems are significantly applied in various domains, including machine learning, signal processing, and economics. Existing algorithms often face challenges such as slow convergence or the inability to guarantee convergence when dealing with such problems. To overcome these limitations, we introduce a dual-relaxed term to enhance the robustness and flexibility of the algorithm. The theoretical analysis shows that the MID-PADMM algorithm can achieve global convergence under suitable conditions and has an iteration complexity of O(1/k), where k represents the number of iterations. Numerical experimental results demonstrate that, compared to the current state-of-the-art algorithms, MID-PADMM exhibits faster convergence rates and higher solution quality in multiple instances.