The alternating direction method of multipliers (ADMM) is a common
optimization tool for solving constrained and non-differentiable problems. We
provide an empirical study of the practical performance of ADMM on several
nonconvex applications, including l0 regularized linear regression, l0
regularized image denoising, phase retrieval, and eigenvector computation. Our
experiments suggest that ADMM performs well on a broad class of non-convex
problems. Moreover, recently proposed adaptive ADMM methods, which
automatically tune penalty parameters as the method runs, can improve algorithm
efficiency and solution quality compared to ADMM with a non-tuned penalty.

本文为非凸优化问题提供一种广泛适用的解决方法，即交替方向乘子法（ADMM），并研究了其在不同领域的实用性，包括线性回归、图像降噪、相位恢复及特征向量计算， 最近从提出的自适应 ADMM 方法可以通过自动调整惩罚参数来提高算法效率和解决质量 compared to ADMM with a non-tuned penalty。

ADMM 在非凸问题中的实证研究

An Empirical Study of ADMM for Nonconvex Problems

The alternating direction method of multipliers (ADMM) is a versatile tool
for solving a wide range of constrained optimization problems, with
differentiable or non-differentiable objective functions. Unfortunately, its
performance is highly sensitive to a penalty parameter, which makes ADMM often
unreliable and hard to automate for a non-expert user. We tackle this weakness
of ADMM by proposing a method to adaptively tune the penalty parameters to
achieve fast convergence. The resulting adaptive ADMM (AADMM) algorithm,
inspired by the successful Barzilai-Borwein spectral method for gradient
descent, yields fast convergence and relative insensitivity to the initial
stepsize and problem scaling.

本研究提出了一种自适应调节罚项参数的方法，使得交替方向乘子法 (ADMM) 得以快速收敛，从而提高了其可靠性和自动化水平。结果表明，这种自适应 ADMM (AADMM) 算法可以相对缓解 ADMM 算法在惩罚因子调节方面存在的问题，并同时保持收敛速度较快。