Stochastic Programming with Primal-Dual Dynamics: A Mean-Field Game Approach

TL;DR

This paper introduces a novel heuristic for stochastic capacity network design using a mean-field game framework, demonstrating convergence to consensus among agents with control over primal-dual dynamics.

Contribution

It formulates the capacity network design as a mean-field game and proves the existence of equilibrium and convergence of consensus dynamics.

Findings

Consensus achieved on network capacity variables

Existence of mean-field equilibrium proven

Control and state penalties influence agent dynamics

Abstract

This study addresses primal-dual dynamics for a stochastic programming problem for capacity network design. It is proven that consensus can be achieved on the \textit{here and now} variables which represent the capacity of the network. The main contribution is a heuristic approach which involves the formulation of the problem as a mean-field game. Every agent in the mean-field game has control over its own primal-dual dynamics and seeks consensus with neighboring agents according to a communication topology. We obtain theoretical results concerning the existence of a mean-field equilibrium. Moreover, we prove that the consensus dynamics converge such that the agents agree on the capacity of their respective micro-networks. Lastly, we emphasize how penalties on control and state influence the dynamics of agents in the mean-field game.