Constraint-coupled Optimization with Unknown Costs: A Distributed Primal Decomposition Approach

TL;DR

This paper introduces a distributed primal decomposition algorithm for convex, constraint-coupled optimization problems where cost functions are unknown and must be learned online, ensuring convergence to the optimal solution.

Contribution

It presents a novel fully distributed method that learns cost functions online and guarantees asymptotic optimality despite unknown costs.

Findings

Algorithm converges to the optimal solution asymptotically.

The method is scalable and preserves agent privacy.

Numerical results validate theoretical claims.

Abstract

In this paper, we present a distributed algorithm for solving convex, constraint-coupled, optimization problems over peer-to-peer networks. We consider a network of processors that aim to cooperatively minimize the sum of local cost functions, subject to individual constraints and to global coupling constraints. The major assumption of this work is that the cost functions are unknown and must be learned online. We propose a fully distributed algorithm, based on a primal decomposition approach, that uses iteratively refined data-driven estimations of the cost functions over the iterations. The algorithm is scalable and maintains private information of agents. We prove that, asymptotically, the distributed algorithm provides the optimal solution of the problem even though the true cost functions are never used within the algorithm. The analysis requires an in-depth exploration of the primal decomposition approach and shows that the distributed algorithm can be thought of as an epsilon-subgradient method applied to a suitable reformulation of the original problem. Finally, numerical computations cor- roborate the theoretical findings and show the efficacy of the proposed approach.