Distributed Convex Optimization with Inequality Constraints over Time-Varying Unbalanced Digraphs

This paper considers a distributed convex optimization problem with inequality constraints over time-varying unbalanced digraphs, where the cost function is a sum of local objective functions, and each node of the graph only knows its local objective and inequality constraints. Although there is a vast body of literature on distributed optimization. most of them require the graph to be balanced, which is quite restrictive and not necessary. To solve it, this work proposes a novel idea of using the epigraph form of the constrained optimization. which can be easily used to study time-varying unbalanced digraphs. Under local communications, a simple iterative algorithm is then designed for each node. We prove that if the graph is uniformly jointly strongly connected, each node asymptotically converges to some common optimal solution.