We investigate the techniques and ideas used in Shefi and Teboulle (SIAM J Optim 24(1), 269–297, 2014) in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration, we allow the use of variable metrics, while the investigations are carried out in the setting of infinite-dimensional Hilbert spaces. This algorithmic scheme is investigated from the point of view of its convergence properties.

我们研究了 Shefi 和 Teboulle (SIAM J Optim 24 (1), 269–297, 2014) 在两个近端 ADMM 算法的收敛分析中使用的技术和思想，这些算法用于解决涉及线性算子组合的凸优化问题。除此之外，我们制定了 ADMM 算法的变体，该算法能够处理目标中涉及附加平滑函数的凸优化问题，并通过其梯度进行评估。此外，在每次迭代中，我们允许使用变量度量，而研究是在无限维希尔伯特空间的设置中进行的。从收敛特性的角度研究了该算法方案。