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A Two-Way Research Street: Geometric Algorithms in Optimization and Computer-Based Discrete Geometry

双向研究街：优化中的几何算法和基于计算机的离散几何

基本信息

批准号：
1818969
负责人：
Jesus De Loera
金额：
$ 30.68万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1818969&HistoricalAwards=false
关键词：
Two Way Research Street Geometric

项目摘要

At the foundation of the progress in artificial intelligence and data science that is changing society (e.g., driverless cars) is the mathematical theory of optimization. For example, convex and non-linear optimization is the engine at the core of the very successful deep neural-networks. The first part of the project develops the mathematics necessary to solve optimization theory challenges (e.g., larger amounts of data, uncertain data) and to create faster, more accurate optimization algorithms. Computers are changing the nature of mathematical research and discovery too. For instance, computers can derive formulas and proofs automaticaly, computers can search for examples, and now they can more easily extract patterns thanks to machine learning. The second part of the project investigates the use of algorithms from artificial intelligence and algorithms to attack problems in mathematics, especially in geometry.This project in computational mathematics has two interacting components: The first component is to apply methods from convex geometry, algebraic geometry, geometry of numbers, and combinatorics to develop new algorithms for mixed-integer optimization problems arising in data science, especially the clustering of data with special conditions. The project also studies augmentation (primal) algorithms for integer and mixed-integer variables, these are algorithms that generalize the pivoting used for the simplex method. The second component of the project investigates geometric and combinatorial problems amenable to be investigated with computers. The computation of a number of fundamental combinatorial quantities in convex geometry, including the exact value of integer Caratheodory numbers for cones, quantitative Helly numbers, and integral Radon-Tverberg numbers, will be emphasized. The project presents a computer-based approach to prove or disprove several theorems indiscrete geometry.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.

在人工智能和数据科学进步的基础上，正在改变社会（例如，无人驾驶汽车）是优化的数学理论。例如，凸优化和非线性优化是非常成功的深度神经网络的核心引擎。该项目的第一部分开发了解决优化理论挑战所需的数学（例如，更大量的数据，不确定的数据），并创建更快，更准确的优化算法。计算机也正在改变数学研究和发现的本质。例如，计算机可以自动推导公式和证明，计算机可以搜索示例，现在由于机器学习，它们可以更容易地提取模式。该项目的第二部分研究使用人工智能算法和算法来解决数学问题，特别是几何问题。计算数学的这个项目有两个相互作用的组成部分：第一个组成部分是应用凸几何，代数几何，数字几何和组合学的方法来开发数据科学中出现的混合整数优化问题的新算法，特别是对具有特殊条件的数据的聚类。该项目还研究了整数和混合整数变量的增广（原始）算法，这些算法推广了单纯形法的旋转。该项目的第二个组成部分调查几何和组合问题，适合用计算机进行调查。在凸几何中的一些基本的组合量的计算，包括整数Caratheodory数的锥，定量Helly数和积分Radon-Tverberg数的精确值，将被强调。该项目提出了一种基于计算机的方法来证明或反驳离散几何中的几个定理。该奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。