Computational Methods for Mathematical Programs with Equilibrium Constraints (MPECs) and Their Economic Applications

从根本上说，经济学研究的是经济行为者在追求自身利益的同时，如何决定组织他们的活动并分配他们的联合生产。 这些相互作用的数学建模需要使用复杂的技术，从数学规划文献，特别是不断增长的领域的数学规划与平衡约束。 标准的优化理论和数值软件假设解满足一个约束条件，而经济和工程中的许多问题并不满足这个假设。 此外，许多问题具有非凸性，这会导致专注于寻找局部解的算法出现问题。 因此，必须开发新的算法来解决现代经济分析中出现的优化问题。 这项研究将开发出比现有技术更强大的新方法来解决这些问题。 为了促进经济学家和其他人的进一步研究，该项目将在公共领域软件中实施新方法，并分发说明性示例，以帮助教用户如何使用该软件。新的数学技术将对许多重要经济问题的研究具有实质性价值。 该项目将研究最优税收问题，其中人口由能力，教育，年龄和家庭状况差异很大的个人组成。 目前的分析着眼于简单得多的例子，由于数学的限制，必须忽略真实的世界的复杂性。 本研究还将展示如何使用这些技术来解决电力市场，医疗保险和其他复杂的经济机构的设计问题。 在这项研究中开发的数值算法将允许许多其他研究人员检查各种各样的问题，目前是不可能使用现有的方法研究。 该奖项是作为2006财政年度数学科学优先领域数学社会和行为科学（MSBS）特别竞赛的一部分获得支持的。

Economics is fundamentally the study of how economic actors decide to organize their activities and to allocate their joint production while each actor is pursuing his own interests. Mathematical modeling of these interactions requires the use of sophisticated techniques from the mathematical programming literature; in particular, the growing field of mathematical programming with equilibrium constraints. Standard optimization theory and numerical software assumes that the solution satisfies a constraint qualification, whereas many problems in economics and engineering do not satisfy this assumption. Moreover, many problems have nonconvexities that cause problems for algorithms focused on finding local solutions. Therefore, new algorithms must be developed to solve the optimization problems arising in modern economic analysis. This study will develop new methods that are more robust than current techniques for solving these problems. To facilitate further research by economists and others, the project will implement the new methods in public domain software and circulate illustrative examples that will help teach users how to use the software.The new mathematical techniques will have substantial value for the study of many important economic problems. The project will study problems in optimal taxation where the population is made up of individuals of widely varying abilities, education, age, and family status. Current analyses look at far simpler examples and must ignore real world complexity due to mathematical limitations. This study also will show how to use these techniques to address problems in the design of electricity markets, health insurance, and other complex economic institutions. The numerical algorithms developed in this study will allow many other researchers to examine a wide variety of problems that currently are impossible to study using existing methods. This award was supported as part of the fiscal year 2006 Mathematical Sciences priority area special competition on Mathematical Social and Behavioral Sciences (MSBS).