Distributed Optimization for Resource Allocation Problems Under Large Delays

In this paper, a distributed resource allocation problem is investigated based on a continuous-time algorithm in the presence of time-varying communication delays. The considered delays are allowed to take both “large” and “small” delay values in an alternating manner. The “small” delays refer to a class of delays that guarantee the exponential convergence to the optimal solution; otherwise, they are called “large” delays. The existence of “large” delays may affect the behavior of the optimal allocation solution, and the existing algorithms with delays may not be applicable to this kind of “large” delays. To address “large” delay cases, a switched algorithm is first proposed based on switching techniques, and then, such an algorithm is modeled as switched delay systems. Second, by adopting the Lyapunov functional theory, the upper bound of delays is first derived to guarantee the exponential convergence to the optimal solution under “small” delays, and then, a delay-dependent sufficient condition is obtained to ensure exponential convergence of the proposed algorithm based on some restrictions on “large” delays. Finally, we adopt an IEEE 118-bus practical example to show the effectiveness of the proposed algorithm.