The Alternating Direction Method of Multipliers (ADMM) is a widely used method for structured convex optimization, and its practical performance depends strongly on the choice of penalty and relaxation parameters. Motivated by settings such as Model Predictive Control (MPC), where one repeatedly solves related optimization problems with fixed structure and changing parameter values, we propose learning online updates of the relaxation parameter to improve performance on problem classes of interest. This choice is computationally attractive in OSQP-like architectures, since adapting relaxation does not trigger the matrix refactorizations associated with penalty updates. We establish convergence guarantees for ADMM with time-varying penalty and relaxation parameters under mild assumptions, and show on benchmark quadratic programs that the resulting learned policies improve both iteration count and wall-clock time over baseline OSQP.

翻译：交替方向乘子法（ADMM）是一种广泛应用于结构化凸优化的方法，其实际性能在很大程度上取决于罚参数和松弛参数的选择。受模型预测控制（MPC）等场景的启发——这些场景中需要反复求解具有固定结构但参数值变化的关联优化问题——我们提出学习松弛参数的在线更新策略，以提升所关注问题类别的性能。该选择在OSQP类架构中具有计算优势，因为调整松弛参数不会触发与罚参数更新相关的矩阵重构。在温和假设条件下，我们建立了时变罚参数和松弛参数下ADMM的收敛保证，并在基准二次规划问题上证明，所学的策略相比基准OSQP在迭代次数和实际运行时间上均有改进。