Program Analysis and Transformation in Mathematical Programming

Abstract

Over the years, mathematical models have become increasingly complex. Rarely can we accurately model a process using only linear or quadratic functions. Instead, we must employ complicated routines written in some programming language. At the same time, most algorithms rely on the ability to exploit structural features within a model. Thus, our ability to compute with a model directly relates to our ability to analyze it. Mathematical programs exemplify these difficult modeling issues. Our desire to accurately model a process is mediated by our ability to solve the resulting problem. Nonetheless, many problems contain hidden structural features that, when identified, allow us to transform the problem into a more computable form. Thus, we must develop methods that not only recognize these hidden features, but exploit them by transforming one problem formulation into another. We present a new domain specific language for mathematical programming. The goal of this language is to develop a system of techniques that allow us to automatically determine the structure of a problem then transform it into a more desirable form. Our technical contribution to this area includes the grammar, type system, and semantics of such a language. Then, we use these tools to develop a series of transformations that manipulate the mathematical model.