Optimization and Equilibria with Expectation Functions: Analysis, Inference and Sampling

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Mathematical optimization and equilibrium problems have prominent applications in machine learning, statistics, economy and business, health care, and many branches of science and engineering. Solving these problems helps to gain knowledge about the nature and structure of complex systems, and to better design and control these systems by making efficient use of scarce resources. There are numerous parameters in the formulation of each such problem. In many cases the exact values of some parameters are not available due to the lack of complete information, especially when such parameters describe future events. An effective way to manage the long-term behaviors of complex systems under such data uncertainty is to introduce probability distributions on the parameters and use expectation functions in the problem formulations. To numerically solve those problems with expectation functions, certain types of approximations are commonly used. The investigators study properties of optimization and equilibrium problems defined with expectation functions, relations between these problems and their approximations, and methods to solve them. Of particular interest is application of these ideas to study the electricity market competition between renewable and nonrenewable energy sources. Results from this project can be used to evaluate the well-posedness of a given problem, measure the reliability of a solution obtained from a numerical procedure, and solve certain types of these problems.The investigators analyze the structure and properties of optimization and equilibrium problems defined by expectation functions, develop inference procedures and sampling-based optimization methods, and study an application to the electricity market. Their first goal is to develop an efficient inference method for the solution to the true optimization or equilibrium problem based on a solution to its sample average approximation (SAA) problem. They expect this method to work for a general framework that allows the SAA functions to be nonsmooth, the SAA solution to be inexact, and the SAA asymptotic distribution to follow a piecewise normal structure as in the cases of general constrained optimization problems. They apply this method to predict the out-of-sample performance of the SAA solutions. Their second goal is to revisit existing importance sampling techniques to efficiently incorporate them into an iterative optimization algorithm for minimizing the probability of a rare event under some design parameters, and study properties of the probability function and compare solutions of the original problem with those of its convex substitutes. The third goal of the investigators is to study a stochastic equilibrium model of the competition behavior between different types of energy generators in the electricity market and to provide a novel generalization of the classical Nash-Cournot equilibrium when the payoff functions are not concave.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.