A Hybrid Multi-Agent System Approach for Distributed Composite Convex Optimization Under Unbalanced Directed Graphs

This paper studies a distributed composite convex optimization problem for multi-agent systems over an unbalanced directed graph. The global objective function is the sum of local cost functions with known mathematical expressions and local cost functions with unknown ones. Due to the particularity of the local cost function, a hybrid multi-agent system composed of continuous-time dynamic agents and discrete-time dynamic agents is employed to solve such a problem. Also, because the local cost function may not be differentiable, a distributed algorithm based on subgradient and gradient-free oracle is proposed. Given some general assumptions, the developed algorithm almost surely converges to an approximately optimal solution. In addition, theoretical analysis indicates that the proposed algorithm possesses the same convergence rate as the existing stochastic gradient-free descent approaches under similar problem settings. Finally, a numerical example is provided to demonstrate the effectiveness of the findings.