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.

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 owing to its superior characteristics, many 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. The flaws are classified into three types, at least one of which exists in each of ADOPT and seven of its variants that we analyzed. In other words, the algorithms may potentially not terminate or terminate with a suboptimal solution. We also propose an amended version of ADOPT that avoids the flaws in existing algorithms and prove that it has the properties of termination and optimality.