We present and analyze a hybrid computational architecture for performing multi-agent optimization. The optimization problems under consideration have convex objective and constraint functions with mild smoothness conditions imposed on them. For such problems, we provide a primal-dual algorithm implemented in the hybrid architecture, which consists of a decentralized network of agents into which an updated dual vector is occasionally injected, and we establish its convergence properties. In this setting, a central cloud computer is responsible for aggregating information, computing dual variable updates, and distributing these updates to the agents. The agents update their (primal) state variables and also communicate among themselves with each agent sharing and receiving state information with some number of its neighbors. Throughout, communications with the cloud are not assumed to be synchronous or instantaneous, and communication delays are explicitly accounted for in the modeling and analysis of the system. Experimental results for a team of robots are presented to support the theoretical developments made.