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交替方向乘子法(Alternating Direction Method of Multipliers,简称ADMM)已经成为求解大规模结构性优化问题的有效方法。尽管已经有较多关于ADMM算法收敛性的研究,但关于该算法参数对收敛性影响的定量表示仍须进一步研究,已有的结果中仅是在实验中凭经验对步长进行选取。文章研究ADMM算法l_1正则化最小的一个重要问题Lasso的收敛因子。研究发现解的形式可用软阈值算子表示,分析发现软阈值的三种情况可以等价转化成算法收敛因子的两种情况,然后通过最小化收敛因子解出最优的步长。实验表明,应用该方法选出的步长,其相应算法的收敛速度明显快于其他选取步长的情况。此外,将该方法应用到压缩感知问题,给出了一个计算最优步长的近似值策略,获得了较好的实验效果。
The Alternating Direction Method of Multipliers (ADMM) has become an effective method to solve large-scale structural optimization problems. Although more researches on the convergence of ADMM algorithms have been made, the quantitative expression of the influence of the parameters on the convergence of the algorithm still needs further study. The existing results only empirically select the step size in experiments. The article studies the convergence factor of Lasso, one of the most important problems in regularization of ADMM algorithm. The study found that the form of the solution can be expressed by soft-threshold operators. The analysis shows that the three cases of soft-threshold can be equivalently transformed into two cases of the algorithm convergence factor, and then the optimal step size can be obtained by minimizing the convergence factor. Experiments show that the convergence rate of the corresponding algorithm is obviously faster than that of the other chosen steps by using this method. In addition, this method is applied to the compression sensing problem, and an approximation strategy for calculating the optimal step size is given. The experimental results are good.