Variational Conditions: Structure and Computation

本项目的重点是研究一类变分条件的数学结构，并设计和分析有效的计算算法来解决它们。所提出的工作的成功将使分析更简单，更有洞察力，更充分地利用问题结构，对广泛的平衡问题进行分析，这些问题具有实质性的实际用途和内在的数学兴趣。它还会产生更好的计算算法，以及证明它们的方法。本研究项目的子课题包括：-通过分析变分条件的参数化版本，更好地了解退化及其影响，目的是阐明特殊结构对问题的敏感性和稳定性行为的影响。-使用近似导数开发新的计算方法，证明其收敛性，并进行计算测试。-改进线性化方法的收敛分析，并将该分析应用于，例如，上面讨论的计算方法。-使用组合对偶方法结合本项目开发的新的稳定性和灵敏度信息分析特殊结构的变分条件。-将在上述研究中获得的知识应用于分析其他有前途的领域，如拟变分不等式。变分条件是平衡问题的数学模型。它们对经济政策研究、农业经济学、结构工程、交通运输、材料科学、博弈论（即冲突模型）等领域的各种问题都很有用。这些问题特别困难的原因之一是，它们模拟了个人或团体不一致行动的情况，实际上经常是在相互冲突的目的下行动。例如，在道路交通平衡问题中，每个司机都试图找到一条最适合他或她的路线，但许多司机独立选择的影响可能会对每个人都不利（例如，在热门路线上造成拥堵）。提出的研究将试图开发更多关于数学模型如何表示这些问题的知识，这些模型的属性是什么，以及如何使用这些属性来计算模型的数值解。有了这些知识，这些模型就可以用来回答“假设”的问题，从而设计出更好的管理方法。例如，对改善交通流量感兴趣的土木工程师可能会使用这些模型来更准确地预测改善现有道路或建造新道路的效果，从而确定如何最好地使用有限的预算来减少交通拥堵。

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.