By enabling the nodes or agents to solve small-sized subproblems to achieve coordination, distributed algorithms are favored by many networked systems for efficient and scalable computation. While for convex problems, substantial distributed algorithms are available, the results for the more broad nonconvex counterparts are extremely lacking. This paper develops a distributed algorithm for a class of nonconvex and nonsmooth problems featured by (i) a nonconvex objective formed by both separate and composite components regarding the decision variables of interconnected agents, (ii) local bounded convex constraints, and (iii) coupled linear constraints. This problem is directly originated from smart buildings and is also broad in other domains. To provide a distributed algorithm with convergence guarantee, we revise the powerful alternating direction method of multiplier (ADMM) method and proposed a proximal ADMM. Specifically, noting that the main difficulty to establish the convergence for the nonconvex and nonsmooth optimization with ADMM is to assume the boundness of dual updates, we propose to update the dual variables in a discounted manner. This leads to the establishment of a so-called sufficiently decreasing and lower bounded Lyapunov function, which is critical to establish the convergence. We prove that the method converges to some approximate stationary points. We besides showcase the efficacy and performance of the method by a numerical example and the concrete application to multi-zone heating, ventilation, and air-conditioning (HVAC) control in smart buildings.

通过使节点或代理能够解决小型子问题以实现协调，分布式算法受到许多网络系统的青睐，以实现高效和可扩展的计算。而对于凸问题，大量分布式算法可用，更广泛的非凸对应物的结果非常缺乏。本文为一类非凸和非光滑问题开发了一种分布式算法，其特点是（i）由独立和复合组件形成的非凸目标，关于互连代理的决策变量，（ii）局部有界凸约束，以及（iii）耦合线性约束。这个问题直接源于智能建筑，在其他领域也很广泛。为了提供一种具有收敛保证的分布式算法，我们修改了强大的乘法器交替方向法（ADMM）方法，并提出了一种近端ADMM。具体来说，注意到使用 ADMM 建立非凸和非平滑优化收敛的主要困难是假设对偶更新的边界，我们建议以折扣方式更新对偶变量。这导致建立所谓的充分递减和下界的Lyapunov 函数，这对于建立收敛至关重要。我们证明了该方法收敛到一些近似的静止点。我们还通过数值示例展示了该方法的有效性和性能，以及在智能建筑中多区域供暖、通风和空调 (HVAC) 控制中的具体应用。