This paper proposes and analyzes an inexact variant of the proximal generalized alternating direction method of multipliers (ADMM) for solving separable linearly constrained convex optimization problems. In this variant, the first subproblem is approximately solved using a relative error condition whereas the second one is assumed to be easy to solve. In many ADMM applications, one of the subproblems has a closed-form solution; for instance, \(\ell _1\) regularized convex composite optimization problems. The proposed method possesses iteration-complexity bounds similar to its exact version. More specifically, it is shown that, for a given tolerance \(\rho >0\), an approximate solution of the Lagrangian system associated to the problem under consideration is obtained in at most \(\mathcal {O}(1/\rho ^2)\) (resp. \(\mathcal {O}(1/\rho )\) in the ergodic case) iterations. Numerical experiments are presented to illustrate the performance of the proposed scheme.

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乘子的不精确近端广义交替方向方法

本文提出并分析了用于解决可分离的线性约束凸优化问题的近端广义交替方向乘数法（ADMM）的不精确变体。在该变体中，使用相对误差条件近似解决了第一个子问题，而假定第二个子问题很容易解决。在许多ADMM应用程序中，子问题之一具有封闭形式的解决方案。例如，\（\ ell _1 \）正则化凸复合优化问题。所提出的方法具有类似于其精确版本的迭代复杂性边界。更具体地，表明，对于给定的公差\（\ rho> 0 \），最多在与所考虑的问题有关的拉格朗日系统的近似解。\（\ mathcal {O}（1 / \ rho ^ 2）\）（分别是\（\ mathcal {O}（1 / \ rho）\）在遍历的情况下）迭代。数值实验表明了该方案的性能。