Learning Over-Relaxation Policies for ADMM with Convergence Guarantees

TL;DR

This paper proposes a method to learn online relaxation parameters for ADMM, improving its efficiency and convergence guarantees in structured convex optimization tasks like MPC.

Contribution

It introduces a learning approach for adaptive relaxation parameters in ADMM with proven convergence guarantees and practical performance improvements.

Findings

Learned policies reduce iteration count in benchmark problems.

Adaptive relaxation improves wall-clock time over baseline OSQP.

Convergence guarantees hold for time-varying parameters.

Abstract

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.