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Collaborative Research: Next-Generation Cutting Planes: Compression, Automation, Diversity, and Computer-Assisted Mathematics

合作研究：下一代切割面：压缩、自动化、多样性和计算机辅助数学

基本信息

批准号：
2012429
负责人：
Yuan Zhou
金额：
$ 17.98万
依托单位：
University of Kentucky Research Foundation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012429&HistoricalAwards=false
关键词：
Collaborative Research Next Generation Cutting

项目摘要

Mixed-integer optimization is a powerful mathematical decision-making technology related to operations research, data sciences, and artificial intelligence. This project considers applications in which high-stake decisions need to be made quickly and account for unknown future event or risk. In such applications, simulation methods and machine learning cannot give sufficient confidence for protecting against the possibility of catastrophic failures. Instead, one requires multi-parametric optimization to precompute responses, certify their safety, and guarantee the level of performance. In this direction, the investigators will study a key component of optimization algorithms called general purpose cutting planes in a novel multi-parametric setting suitable for process control in chemical engineering and optimizing compilers for high-performance computing platforms, aiming for major theoretical and computational advances that will generalize to many important applications. Broader impacts include the training of undergraduate and graduate students in computational mathematics and research skills, as well as development of high-quality open-source research software, and of further connections between several research communities within mathematics, computer science, and engineering.Mixed-integer (linear and nonlinear) optimization is concerned with finite-dimensional, non-convex optimization problems that include discrete decision variables such as those that model "yes/no" decisions. Systems of this type arise in all areas of industry and the sciences. Algorithms for mixed-integer optimization build upon convex optimization technology by relaxation, approximation, convexification, and decomposition techniques. Increases in system size in the presence of Big Data technologies creates new challenges that need to be addressed by a next generation of algorithms. This project studies convexification, specifically, cutting planes in multi-row and multi-cut cutting plane systems that are effective and efficient from the aspects of compression, automation, and diversity. In particular, spaces of extreme continuous piecewise linear cut-generating functions with prescribed features will be computed; these consist of semi-algebraic cells, parametrizing sub-additive piecewise linear functions, glued at their boundaries. The computation of each cell requires the proof of a theorem, and automated theorem proving technology, based on metaprogramming and semi-algebraic computations, will be developed. The investigators will apply the new cutting plane techniques to two target applications for which guaranteed correctness and performance is mission-critical: model predictive control in chemical process engineering and optimizing compilers for high-performance computing platforms. The multi-parametric optimization problems in both applications will benefit from the parametric nature of the new cutting planes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

混合整数优化是一种与运筹学、数据科学和人工智能相关的强大的数学决策技术。此项目考虑需要快速做出高风险决策并考虑未知未来事件或风险的应用程序。在这样的应用中，模拟方法和机器学习不能为防范灾难性故障的可能性提供足够的信心。取而代之的是，需要多参数优化来预计算响应、证明其安全性并保证性能水平。在这个方向上，研究人员将在一种新的多参数设置中研究优化算法的一个关键组件，称为通用切割平面，该设置适合于化学工程中的过程控制和高性能计算平台的优化编译器，旨在取得重大的理论和计算进展，这些进展将推广到许多重要的应用。更广泛的影响包括本科生和研究生在计算数学和研究技能方面的培训，以及高质量开源研究软件的开发，以及数学、计算机科学和工程领域内几个研究社区之间的进一步联系。混合整数(线性和非线性)优化涉及有限维、非凸优化问题，其中包括离散决策变量，如那些模拟是/否决策的变量。这种类型的系统出现在工业和科学的所有领域。混合整数优化算法通过松弛、逼近、凸化和分解技术建立在凸优化技术的基础上。随着大数据技术的出现，系统规模的增加带来了新的挑战，需要下一代算法来解决。本课题从压缩、自动化、多样性等方面研究凸化问题，即多排多割平面系统中有效且高效的割平面问题。具体地说，将计算具有指定特征的极连续分段线性割生成函数的空间；这些空间由半代数单元、参数化次可加分段线性函数、粘在其边界处的单元组成。每个单元的计算需要一个定理的证明，并且将开发基于元编程和半代数计算的自动定理证明技术。研究人员将把新的切割平面技术应用于两个目标应用程序，对它们来说，保证正确性和性能是关键任务：化学过程工程中的模型预测控制和高性能计算平台的优化编译器。这两个应用中的多参数优化问题将受益于新切割平面的参数性质。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。