Distributed online convex optimization with multiple coupled constraints: A double accelerated push–pull algorithm

Abstract

This paper is concerned with the distributed online convex optimization problems with multiple coupled constraints over unbalanced digraphs, which are widely applied in diverse fields such as machine learning, smart grids, and resource allocation problems, etc. In such practical scenarios, multiple coupled constraints, where each coupled constraint might only contain a portion of all nodes, are more general and complex than global coupled constraints. Additionally, due to certain constraint conditions such as energy and communication limitations, each node can only exchange information with other nodes in an identical constraint. Besides, the network connectivity of practical constrained problems might not be only confined to one type of weight matrices. By adopting an augmented Lagrangian method, the multiple coupled constrained optimization problem is converted into a saddle-point problem. To tackle this problem, a novel double accelerated push–pull (DAPP) algorithm is proposed which concurrently employs row-stochastic (RS) and column-stochastic (CS) weight matrices. Specifically, each node pushes gradient information weighted by CS matrices to out-neighbors, while pulls dual variables fused by RS matrices from in-neighbors. Combined with CS matrices, the gradient tracking scheme is employed to promote the efficiency of tracking the averaged gradients. Furthermore, aiming at improving the convergence rate, the heavy-ball method is used to update both primal and dual variables. Then, rigorous theoretical results indicate that DAPP obtains the sublinear bounds of regret and constraint violation. Finally, the plug-in electric vehicles (PEVs) charging problem is utilized to demonstrate the validity of the proposed algorithm.