Four Mathematical Programming Paradigms with Operations Research Applications

本提案旨在研究数学规划中四种新兴的建模范式，这些范式近年来由于在运筹学和其他工程领域以及经济和金融领域的广泛应用而受到越来越多的关注，但尚未得到广泛的普及和缺乏重点研究。这些范例是：竞争、动态、等级和逆问题。该项目的目标是为这些范例在复杂工程和经济系统背景下的持续处理和应用建立一个全面而严谨的基础。提出的方法是从扩展数学和平衡规划、约束连续时间动力系统、多智能体和多层次优化等方面研究这些范式的数学表达式。新的数学理论和先进的计算工具将发展。这将需要结合连续和析取规划、全局优化、连续和离散时间动力系统理论以及变分和集值分析等当代数学工具的方法。如果成功，该研究结果将有助于提高对高度复杂的工程、经济和生物系统的理解和有效解决方案，这些系统具有四个内在要素：多个自私主体之间的竞争，达到最优或平衡之前的时间进化，优化主体的层次结构，以及需要尊重的历史和/或观察数据。由于其跨学科性质，提议的活动提供了一个机会，汇集了不同学科的专家，以推进他们各自的领域，并共同为重大的社会问题做出贡献，如工程和经济系统中的有效资源分配，交通系统中的拥堵，以及解决竞争自私主体之间的冲突，仅举几个应用领域。

This proposal aims at investigating four emerging modeling paradigmsin mathematical programming that have gained increased attention in recent years due to their pervasive applications in operations research and other engineering fields as well as in economics and finance, but which have yet to receive wide popularity and lack focused research. These paradigms are: competition, dynamics, hierarchy, and inverse problems. The objective of the project is to develop a comprehensive and rigorous foundation for the sustained treatment and applications of these paradigms in the context of complex engineering and economic systems. The proposed approach is to study the mathematical formulations of these paradigms in terms of extended mathematical and equilibrium programs, constrained continuous-time dynamical systems, and multi-agent and multi-level optimization. Novel mathematical theories and advanced computational tools will be developed. This will require the combined methodology of continuous and disjunctive programming, global optimization, continuous- and discrete-time dynamical systems theory, and contemporary mathematical tools such as variational and set-valued analysis.If successful, the results of the proposed research will lead to improved understanding and efficient solution of highly complex engineering, economic, and biological systems characterized by four intrinsic elements: competition among multiple selfish agents, time evolution prior to the attainment of an optimum or equilibrium, hierarchical structure of the optimizing agents, and historical and/or observed data that need to be respected. Due to its interdisciplinary nature, the proposed activity offers an opportunity to bring together experts in diverse disciplines to advance their individual fields and make joint contributions to significant societal problems such as efficient resource allocation in engineering and economic systems, congestion in transportation systems, and resolution of conflicts among competing selfish agents, to name a few areas of applications.