Robustification of Continuous-Time ADMM against Communication Delays under Non-Strict Convexity: A Passivity-Based Approach

Abstract

In this paper, we address a class of distributed optimization problems with non-strictly convex cost functions in the presence of communication delays between an agent and a coordinator. To this end, we focus on a continuous-time optimization algorithm that mirrors the alternating direction method of multipliers. We first redesign the algorithm so that the dynamics ensures smoothness and a sub-block for primal optimization includes stable zeros. It is then revealed that the algorithm is composed of feedback interconnection of passive systems. We next robustify the algorithm against communication delays by applying the so-called scattering transformation. The smoothness of the dynamics allows one to use the invariance principle for delay systems, and accordingly, the state trajectories are shown to converge to an optimal solution even without the strict convexity assumption. Finally, the presented method is demonstrated via simulation of an environmental-monitoring problem.

Keywords

• convex optimization
• alternating direction method of multipliers
• communication delays
• passivity