We propose a novel, system theoretic analysis of the Alternating Direction Method of Multipliers (ADMM) applied to a convex constraint-coupled optimization problem. The resulting algorithm can be interpreted as a linear, discrete-time dynamical system (modeling the multiplier ascent update) in closed loop with a static nonlinearity (representing the minimization of the augmented Lagrangian). When expressed in suitable coordinates, we prove that the discrete-time linear dynamical system has a discrete positive-real transfer function and is interconnected in closed loop with a static, passive nonlinearity. This readily shows that the origin is a stable equilibrium for the feedback interconnection. Finally, we also show global asymptotic stability of the origin for the closed-loop system and, thus, global asymptotic convergence of ADMM to the optimal solution of the optimization problem.

我们对应用于凸约束耦合优化问题的乘法器交替方向法 (ADMM) 提出了一种新颖的系统理论分析。生成的算法可以解释为具有静态非线性（表示增强拉格朗日函数的最小化）的闭环中的线性离散时间动态系统（对乘数上升更新建模）。当用合适的坐标表示时，我们证明了离散时间线性动力系统具有离散的正实传递函数，并且在闭环中以静态、无源非线性相互连接。这很容易表明原点是一个稳定的平衡用于反馈互连。最后，我们还展示了闭环系统原点的全局渐近稳定性，因此，ADMM 对优化问题的最优解的全局渐近收敛。