Polyhedral Combinatorics and Algorithms for Stochastic Integer Programming

这项拨款为研究多面体组合学和随机整数规划（IP）算法提供资金。在过去的十年中，随机IP作为建模和解决考虑不确定事件的实时决策问题的有效工具受到了文献的广泛关注。同时，随机IP结合了整数规划和随机线性规划的复杂性，这给大规模问题的求解带来了挑战。本研究的重点是研究一般随机IP的基本结构和开发求解大规模问题的快速算法。研究工作包括：1)研究随机批量问题的强有效不等式并开发高效的分支切算法；2)探索一般随机IP问题的多面体组合；3)确定几种特殊类型随机IP问题的快速多项式时间和近似算法的性质。我们还将大力开发整数规划和随机优化的教学模块。本科生和研究生，强调未被充分代表的群体，将参与该项目。如果成功，本研究结果将导致随机知识产权的科学方法论创新。多面体研究将有助于开发改进的商业软件，以解决广泛的随机知识产权问题。结果还可以与分解算法和基于优化的启发式相结合，以提高大规模实际问题的建模和计算能力。例子包括生产计划、制造维修、检修、劳动力调度和设备位置问题。研究成果将最终提供可纳入研究生水平课程的内容和方法，并可作为开发新课程的基础。

This grant provides funding to study polyhedral combinatorics and algorithms for stochastic integer programming (IP). During the last decade, stochastic IP has received broad attention in the literature as an efficient tool for modeling and solving real-time decision making problems with the consideration of uncertain events. Meanwhile, stochastic IP incorporates both the complexity of integer programming and stochastic linear programming, which makes it challenging to solve large-scale problems. This research focuses on studying fundamental structures of general stochastic IP and developing fast algorithms for the solution of large-scale problems. The research work consists of 1) studying strong valid inequalities and developing efficient branch-and-cut algorithms for stochastic lot-sizing problems, 2) exploring polyhedral combinatorics for general stochastic IP, and 3) identifying properties that would lead to fast polynomial time and approximation algorithms for several special classes of stochastic IP problems. Significant efforts will also be spent developing teaching modules for integer programming and stochastic optimization. Undergraduate and graduate students, emphasizing underrepresented groups, will participate in the project.If successful, the results of this research will lead to scientific methodology innovations for stochastic IP. Polyhedral studies will contribute to the development of improved commercial software for a wide range of stochastic IP problems. The results may also be combined with decomposition algorithms and optimization-based heuristics to improve modeling and computational capabilities on large-scale practical problems. Examples include production planning, manufacturing repair overhaul workforce scheduling, and facility location problems. The research outcomes will finally provide content and methodologies that can be incorporated into graduate level courses and can serve as the bases for development of new courses.