This project is focused on investigating the mathematical structure of certain classes of variational conditions, and designing and analyzing effective computational algorithms to solve them. Success in the proposed work would permit analysis to be done more simply, with better insight, and with greater exploitation of problem structure, for a broad class of equilibrium problems having substantial practical usefulness as well as intrinsic mathematical interest. It would also produce better computational algorithms, and the means for justifying them. Sub-topics in this research program include:
- Gaining better insight into degeneracy and its effects through analysis of parameterized versions of variational conditions, with the aim of illuminating the effect of special structure on the sensitivity and stability behavior of the problem.
- Developing new computational methods using approximate derivatives, justifying their convergence, and testing them computationally.
- Improving convergence analysis for methods of linearization type, and applying that analysis to, e.g., the computational methods discussed just above.
- Analyzing specially structured variational conditions using methods of composition duality combined with the new stability and sensitivity information developed in this project.
- Applying knowledge gained in the above research to analysis of other promising areas such as quasi-variational inequalities.
Variational conditions are mathematical models of equilibrium problems. They are useful for a wide variety of problems from areas including economic policy studies, agricultural economics, structural engineering, transportation, materials science, game theory (that is, models of conflict), and others. One of the reasons why these problems are especially hard is that they model situations in which individuals or groups are not acting in concert, and indeed are often acting at cross purposes. For example, in a road traffic equilibrium problem each driver is trying to find a route that is best for him or her, but the effects of many drivers' independent choices can make conditions bad for everyone (e.g., by creating congestion on popular routes). The research that is proposed will try to develop more knowledge about how mathematical models can represent these problems, what the properties of those models are, and how to use those properties to compute numerical solutions to the models. With such knowledge the models could then be used to answer "what-if" questions and thus to design better management methods. For example, civil engineers interested in improving traffic flow might use such models to predict more accurately the effect of improving existing roads or of building new roads, and thereby determine how best to spend a limited budget to reduce traffic congestion.
