Constrained Distributed Nonconvex Optimization over Time-varying Directed Graphs

This paper addresses a class of constrained distributed nonconvex optimization problems involving univariate objective functions, considering time-varying directed networks. We propose a novel algorithm named Extended-CPCA (E-CPCA), exploiting the notion of combining Chebyshev polynomial approximation and average consensus. The proposed algorithm is i) able to yield epsilon globally optimal solutions for any arbitrarily small given tolerance epsilon, ii) efficient in terms of both oracle complexities and inter-agent communication costs, and iii) distributed terminable when the specified precision requirement is met. The idea of leveraging polynomial proxy and consensus to deal with the mentioned problems over static undirected graphs is first presented in our previous work. The novelties of this work lie in i) the utilization of push-sum average consensus with distributed stopping mechanism to enable agents to acquire a proxy for the global objective over time-varying digraphs without much wastes of extra communications, and ii) the transformation of the optimization of this global proxy into a semidefinite program to help in obtaining solutions in a fast and reliable manner. Both the analysis and simulations are provided to illustrate the efficacy of the proposed algorithm.