A distributed constraint optimization problem (DCOP) is a framework to model multi-agent coordination problems. Asynchronous distributed optimization (ADOPT) is a well-known complete DCOP algorithm, and many of its variants have been proposed over the last decade. It is considered proven that ADOPT-based algorithms have the key properties of termination and optimality, which guarantee that the algorithms terminate in a finite time and obtain an optimal solution, respectively. In this paper, we present counterexamples to the termination and optimality of ADOPT-based algorithms. They are classified into three types, at least one of which exists in each of ADOPT and eight of its variants that we analyzed. In other words, the algorithms may potentially not terminate or terminate with a suboptimal solution. Furthermore, we show that the bounded-error approximation of ADOPT, which enables the algorithm to terminate faster with the quality of the solution guaranteed within a predefined error bound, also suffers from flaws. Additionally, we propose an amended version of ADOPT that avoids the flaws in existing algorithms and prove that it has the properties of termination and optimality.

分布式约束优化问题（DCOP）是对多智能体协调问题进行建模的框架。异步分布式优化 (ADOPT) 是一种众所周知的完整 DCOP 算法，在过去十年中已经提出了它的许多变体。事实证明，基于ADOPT的算法具有终止性和最优性的关键属性，分别保证算法在有限时间内终止并获得最优解。在本文中，我们提出了基于 ADOPT 的算法的终止和最优性的反例。它们分为三种类型，其中至少一种存在于我们分析的每个 ADOPT 及其八个变体中。换句话说，算法可能不会终止或以次优解决方案终止。此外，我们表明，ADOPT 的有界误差近似也存在缺陷，该近似使算法能够更快地终止，并在预定义的误差范围内保证解的质量。此外，我们提出了ADOPT的修正版本，避免了现有算法的缺陷，并证明它具有终止性和最优性。