CPCA: A Chebyshev Proxy and Consensus based Algorithm for General Distributed Optimization

We consider a general distributed optimization problem, aiming to optimize the average of a set of local objectives that are Lipschitz continuous univariate functions, with the existence of same local constraint sets. To solve the problem, we propose a Chebyshev Proxy and Consensus-based Algorithm (CPCA). Compared with existing distributed optimization algorithms, CPCA is able to address the problem with non-convex Lipschitz objectives, and has low computational costs since it is free from gradient or projection calculations. These benefits result from i) the idea of optimizing a Chebyshev polynomial approximation (i.e. a proxy) for the global objective to obtain (∈)-optimal solutions for any given precision (∈), and ii) the use of average consensus where the local proxies' coefficient vectors are gossiped to enable every agent to obtain such a global proxy. We provide comprehensive analysis of the accuracy and complexities of the proposed algorithm. Simulations are conducted to illustrate its effectiveness.