Sun Min, Liu Jing
School of Management, Qufu Normal University, Shandong, 276826 P.R. China ; School of Mathematics and Statistics, Zaozhuang University, Shandong, 277160 P.R. China.
School of Data Sciences, Zhejiang University of Finance and Economics, Hangzhou, 310018 China.
J Inequal Appl. 2017;2017(1):19. doi: 10.1186/s13660-017-1295-1. Epub 2017 Jan 14.
The proximal alternating direction method of multipliers (P-ADMM) is an efficient first-order method for solving the separable convex minimization problems. Recently, He have further studied the P-ADMM and relaxed the proximal regularization matrix of its second subproblem to be indefinite. This is especially significant in practical applications since the indefinite proximal matrix can result in a larger step size for the corresponding subproblem and thus can often accelerate the overall convergence speed of the P-ADMM. In this paper, without the assumptions that the feasible set of the studied problem is bounded or the objective function's component [Formula: see text] of the studied problem is strongly convex, we prove the worst-case [Formula: see text] convergence rate in an ergodic sense of the P-ADMM with a general Glowinski relaxation factor [Formula: see text], which is a supplement of the previously known results in this area. Furthermore, some numerical results on compressive sensing are reported to illustrate the effectiveness of the P-ADMM with indefinite proximal regularization.
近端乘子交替方向法（P-ADMM）是一种求解可分离凸最小化问题的高效一阶方法。最近，何进一步研究了P-ADMM，并将其第二个子问题的近端正则化矩阵放宽为不定矩阵。这在实际应用中尤为重要，因为不定近端矩阵可以为相应子问题带来更大的步长，从而通常可以加快P-ADMM的整体收敛速度。在本文中，在不假设所研究问题的可行集有界或所研究问题的目标函数分量[公式：见原文]是强凸的情况下，我们证明了具有一般Glowinski松弛因子[公式：见原文]的P-ADMM在遍历意义下的最坏情况[公式：见原文]收敛速度，这是该领域先前已知结果的补充。此外，还报告了一些关于压缩感知的数值结果，以说明具有不定近端正则化的P-ADMM的有效性。
