Distributed and parallel algorithms have been frequently investigated in the recent years, in particular in applications like machine learning. Nonetheless, only a small subclass of the optimization algorithms in the literature can be easily distributed, for the presence, e.g., of coupling constraints that make all the variables dependent from each other with respect to the feasible set. Augmented Lagrangian methods are among the most used techniques to get rid of the coupling constraints issue, namely by moving such constraints to the objective function in a structured, well-studied manner. Unfortunately, standard augmented Lagrangian methods need the solution of a nested problem by needing to (at least inexactly) solve a subproblem at each iteration, therefore leading to potential inefficiency of the algorithm. To fill this gap, we propose an augmented Lagrangian method to solve convex problems with linear coupling constraints that can be distributed and requires a single gradient projection step at every iteration. We give a formal convergence proof to at least \(\varepsilon \)-approximate solutions of the problem and a detailed analysis of how the parameters of the algorithm influence the value of the approximating parameter \(\varepsilon \). Furthermore, we introduce a distributed version of the algorithm allowing to partition the data and perform the distribution of the computation in a parallel fashion.

近年来，尤其在诸如机器学习之类的应用中，分布式和并行算法已得到广泛研究。但是，由于存在例如耦合约束，使得所有变量相对于可行集相互依赖，因此，文献中的优化算法只有很小的子类可以轻松分配。增强拉格朗日方法是摆脱耦合约束问题的最常用技术之一，即通过以结构化，深入研究的方式将此类约束移至目标函数。不幸的是，标准的扩充拉格朗日方法需要解决嵌套问题，因为每次迭代需要（至少不精确地）解决一个子问题，因此导致算法的潜在效率低下。为了填补这一空白，我们提出了一种增强的拉格朗日方法来解决线性耦合约束的凸问题，该线性耦合约束可以分布并且每次迭代都需要一个梯度投影步骤。我们至少可以提供正式的收敛证明\（\ varepsilon \） -问题的近似解，并详细分析算法的参数如何影响近似参数\（\ varepsilon \）的值。此外，我们介绍了该算法的分布式版本，该算法允许对数据进行分区并以并行方式执行计算的分配。