Optimization and Control of Large Systems: Fighting the Curse of Dimensionality

Overview & Synopsis

The goal of this thesis is to provide methods for the control and the optimization of large-scale systems, starting from stochastic models that approximate its behavior. However, such models suffer from the curse of dimensionality: the number of states needed to represent a system explodes when the size of the system grows. In a first part, we present two different methods to reduce the complexity of the model by aggregating the states of the different objects. These two methods are illustrated by analyzing the performance of a distributed load balancing strategy, namely work-stealing. We first show how the use of a potential function leads to a tight analysis of the total completion time of a bag of tasks. Then we show how a mean field approximation can be used to study the steady-state of a grid scheduled by work-stealing. Then, we focus on the optimal control of large stochastic systems. We extend classical mean field methods to study the controlled behavior of large systems. We show that under mild assumptions, solving an optimal control problem for a system with a large number of objects can be reduced to the solving of a problem for a deterministic system as the number of objects grows large. In practice, this allows one to evaluate the performance of a policy and provide a way to asymptotically solve problems that used to be intractable. The last part of the document studies two different problems. We first consider the question of how to distribute a infinite number of tasks on an infinite tree, as evenly as possible. Secondly we study a distributed power control algorithm in wireless network whose goal is to efficiently balance the trade-off between the delay and the power spent per transmitted packet.