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Distributed Optimization on Unbalanced Time-Varying Topologies: Theories and Experiments

This paper considers distributed optimization problems on both fixed and time-varying directed communication structures within a continuous-time framework. Different from most existing results in literature on continuous-time optimization, the lower bound of local convexity constants are unknown, and the requirement of weight-balanced communication structures is removed. An augmented Lagrangian function is designed to analyze the properties of optimal solutions over an asymmetric Laplacian matrix. Based on this analysis, a consensus-based algorithm is proposed to solve the distributed optimization problem on unbalanced directed graphs such that the algorithm asymptotically converges to optimal solutions. Furthermore, an algorithm is proposed to solve the distributed optimization problem on unbalanced time-varying communication topologies. A novel Lyapunov function including a semi-positive definite term is designed to establish the convergence analysis of this algorithm. By exploring certain features of positive invariance sets and asymmetric Laplacian matrices, sufficient conditions for the convergence are established. Two experiments are carried out on a distributed microcomputer platform, thus validating the proposed algorithms.