Social Optima in Leader-Follower Mean Field Linear Quadratic Control

This paper investigates a linear quadratic mean field leader-follower team problem, where the model involves one leader and a large number of weakly-coupled interactive followers. The leader and the followers cooperate to optimize the social cost. Specifically, for any strategy provided first by the leader, the followers would like to choose a strategy to minimize social cost functional. Using variational analysis and person-by-person optimality, we construct two auxiliary control problems. By solving sequentially the auxiliary control problems with consistent mean field approximations, we can obtain a set of decentralized social optimality strategy with help of a class of forward-backward consistency systems. The relevant Stackelberg equilibrium is further proved under some proper conditions

A Deterministic Annealing Framework for Global Optimization of Delay-Constrained Communication and Control Strategies

This dissertation is concerned with the problem of global optimization of delay constrained communication and control strategies. Specifically, the objective is to obtain optimal encoder and decoder functions that map between the source space and the channel space, to minimize a given cost functional. The cost surfaces associated with these problems are highly complex and riddled with local minima, rendering gradient descent based methods ineffective. This thesis proposes and develops a powerful non-convex optimization method based on the concept of deterministic annealing (DA) - which is derived from information theoretic principles with analogies to statistical physics, and was successfully employed in several problems including vector quantization, classification and regression. DA has several useful properties including reduced sensitivity to initialization and strong potential to avoid poor local minima. DA-based optimization methods are developed here for the following fundamental communication problems: the Wyner-Ziv setting where only a decoder has access to side information, the distributed setting where independent encoders transmit over independent channels to a central decoder, and analog multiple descriptions setting which is an extension of the well known source coding problem of multiple descriptions. Comparative numerical results are presented, which show strict superiority of the proposed method over gradient descent based optimization methods as well as prior approaches in literature. Detailed analysis of the highly non-trivial structure of obtained mappings is provided. The thesis further studies the related problem of global optimization of controller mappings in decentralized stochastic control problems, including Witsenhausen's celebrated 1968 counter-example. It is well-known that most decentralized control problems do not admit closed-form solutions and require numerical optimization. An optimization method is developed, based on DA, for a class of decentralized stochastic control problems. Comparative numerical results are presented for two test problems that show strict superiority of the proposed method over prior approaches in literature, and analyze the structure of obtained controller functions

Suboptimal Solutions to Team Optimization Problems with Stochastic Information Structure

Existence, uniqueness, and approximations of smooth solutions to team optimization problems with stochastic information structure are investigated. Suboptimal strategies made up of linear combinations of basis functions containing adjustable parameters are considered. Estimates of their accuracies are derived by combining properties of the unknown optimal strategies with tools from nonlinear approximation theory. The estimates are obtained for basis functions corresponding to sinusoids with variable frequencies and phases, Gaussians with variable centers and widths, and sigmoidal ridge functions. The theoretical results are applied to a problem of optimal production in a multidivisional firm, for which numerical simulations are presented