The theory of mean field games has been developed in the last decade by economists, engineers, and mathematicians in order to study decision making in very large populations of small interacting agents. The approach by Lasry and Lions leads to a system of nonlinear partial differential equations, the solution of which can be used to approximate the limit of an N-player Nash equilibrium as N tends to infinity. This talk will mainly focus on deterministic models, which are associated with a first order pde system. The main points that will be addressed are the existence and uniqueness of solutions, their regularity in the presence of boundary conditions, and their asymptotic behaviour as time goes to infinity.
One of the most important developments in the design of optimization algorithms came with the advent of quasi-Newton methods in the 1960s. These methods have been used to great effect over the past decades, and their heyday is not yet over. In this talk, we introduce a new advance in quasi-Newton methodology that we refer to as displacement aggregation. We also discuss other new advances for extending quasi-Newton ideas for stochastic and nonconvex, nonsmooth optimization.