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

经济学从根本上讲是研究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和分配他们的共同生产。这些相互作用的数学建模需要使用数学规划文献中的复杂技术；特别是具有平衡约束的数学规划领域。标准的优化理论和数值软件假定解满足约束条件，而经济和工程中的许多问题并不满足这一假设。此外，许多问题具有非凸性，这给专注于寻找局部解的算法带来了问题。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目前解决这些问题的技术更强大的新方法。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的软件中实施新方法，并传播说明性示例，以帮助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的价值。该项目将研究在人口由能力、教育、年龄和家庭状况差别很大的个人组成的情况下的最优税收问题。当前的分析着眼于更简单的例子，由于数学的限制，必须忽略现实世界的复杂性。本研究还将展示如何使用这些技术来解决电力市场、健康保险和其他复杂经济制度设计中的问题。在这项研究中开发的数值算法将使许多其他研究人员能够检查目前使用现有方法无法研究的各种问题。该奖项是2006财政年度数学科学优先领域数学社会和行为科学特别竞赛的一部分。

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).