Mathematical programs with equilibrium constraints: automatic reformulation and solution via constrained optimization

Constrained optimization has been extensively used to solve many large scale deterministic problems arising in economics, including, for example, square systems of equations and nonlinear programs. A separate set of models have been generated more recently, using complementarity to model various phenomenon, particularly in general equilibria. The unifying framework of mathematical programs with equilibrium constraints (MPEC) has been postulated for problems that combine facets of optimization and complementarity. This paper briefly reviews some methods available to solve these problems and described a new suite of tools for working with MPEC models. Computational results demonstrating the potential of this tool are given that automatically construct and solve a variety of different nonlinear programming reformulations of MPEC problems.\ud \ud This material is based on research partially supported by the National Science Foundation Grant CCR-9972372, the Air Force Office of Scientific Research Grant F49620-01-1-0040, Microsoft Corporation and the Guggenheim Foundation

Neuro-Dynamic Programming for Radiation Treatment Planning

In many cases a radiotherapy treatment is delivered as a series of smaller dosages over a period of time. Currently, it is difficult to determine the actual dose that has been delivered at each stage, precluding the use of adaptive treatment plans. However, new generations of machines will give more accurate information of actual dose delivered, allowing a planner to compensate for errors in delivery. We formulate a model of the day-to-day planning problem as a stochastic linear program and exhibit the gains that can be achieved by incorporating uncertainty about errors during treatment into the planning process. Due to size and time restrictions, the model becomes intractable for realistic instances. We show how neuro-dynamic programming can be used to approximate the stochastic solution, and derive results from our models for realistic time periods. These results allow us to generate practical rules of thumb that can be immediately implemented in current planning technologies.\ud \ud This material is based on research partially supported by the National Science Foundation Grants ACI-0113051 and CCR-9972372, the Air Force Office of Scientific Research Grant F49620-01-1-0040, Microsoft Corporation and the Guggenheim Foundation

Limited memory solution of complementarity problems arising in video games

We describe the solution of a complementarity problem with limited memory resources. The problem arising from physical simulations occurring within video games. The motivating problem is outlined, along with a simple interior point approach for its solution. Various linear algebra issues arising in the implementation are explored, including preconditioning, ordering and various ways of solving an equivalent augmented system. Alternative approaches are briefly surveyed, and some recommendations for solving these types of problem are given.\ud \ud This material is based on research partially supported by the Smith Institute, EPSRC Grant GR/M59044, the National Science Foundation Grant CCR-9972372, the Air Force Office of Scientific Research Grant F49620-01-1-0040, and the Guggenheim Foundation