Four Mathematical Programming Paradigms with Operations Research Applications

这一建议旨在研究数学规划中的四种新兴建模范例，它们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近年来受到越来越多的关注，但尚未得到广泛的普及和重点研究。这些范例是：竞争、动力学、层级和逆问题。该项目的目标是为在复杂的工程和经济系统中持续处理和应用这些范例奠定全面和严格的基础。所提出的方法是根据扩展的数学和平衡规划、受约束的连续时间动态系统以及多智能体和多水平优化来研究这些范例的数学公式。新的数学理论和先进的计算工具将被开发出来。这将需要连续和析取规划、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现代数学工具的组合方法论。如果成功，拟议的研究结果将导致对高度复杂的工程、经济和生物系统的更好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，达到最优或均衡之前的时间演化，优化主体的层次结构，以及需要尊重的历史和/或观测数据。由于其跨学科性质，拟议的活动提供了一个机会，将不同学科的专家聚集在一起，推进各自的领域，共同为重大的社会问题作出贡献，如工程和经济系统中的有效资源分配、交通系统的拥堵，以及解决相互竞争的自私代理人之间的冲突，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.