Stochastic variational inequalities model a large number of equilibrium problems subject to data uncertainty. They are closely related to stochastic optimization problems, which form an important subfield of mathematical optimization. The investigator and her colleagues study the mathematical structure of stochastic variational inequalities, develop their solution methods and analytic tools, and apply these to handle data uncertainty in equilibrium and optimization problems. They analyze the relation between parameters and solutions of stochastic variational inequalities, investigate the convergence of major computational methods such as the sample average approximation method, and use such knowledge to develop and justify methods to establish confidence regions for solutions to stochastic variational inequalities and stochastic optimization problems. To broaden the application areas of these methods, they extend the work from single-stage problems to multi-stage ones, and from the single measure of expected values to a general class of risk measures. They implement the resulting computational tools in specific operational planning problems.
The variational inequality is a mathematical problem that characterizes an equilibrium status. It provides a unified framework for a very large number of problems from areas including materials science, transportation, computer networks, signal processing, mechanics, economics, and other areas. Whereas deterministic variational inequalities are formulated with fixed parameters, real world problems almost invariably include some uncertain parameters. Such uncertainty may be caused by lack of reliable data, measurement errors, future and unobservable events, and so on. A common approach to tackling such uncertainty is by introducing a probability distribution on the parameters. The resulting problem is a stochastic variational inequality, which aims to find the "average" equilibrium status. Stochastic variational inequalities have been used to model a wide variety of problems in energy market analysis, communication network modeling, transportation planning, inventory management, and other areas. In this research, the investigator and her colleagues study to develop better insight into the mathematical structure of stochastic variational inequalities, and to exploit this insight to produce better computational algorithms and analytic tools. They investigate how solutions to these problems depend on their data, how to find a good solution effectively, and how to identify practical problems that are easy to solve and have robust behavior. Success in the work helps to better understand, analyze and solve a broad class of equilibrium and optimization problems subject to data uncertainty.