CAREER: Core-Sets for Geometric Optimization and their Applications.

自核心集被发现以来，在非常短的时间内，它已经成为研究几何优化问题的有力工具。该技术背后的主要思想是计算输入的一个小子集，称为核心集，它“近似”输入，并允许在这个简洁的表示上解决优化问题。核心集是令人兴奋的，因为它们在开发强大、实用和通用的方法方面具有广阔的前景，这些方法适用于各种应用中的大量问题，包括汽车和飞机工业、空间探索、土木测量师、电影、电脑游戏和医疗行业，以及安全和地理信息系统的开发人员。这份NSF职业计划书的重点是设计和实现几何优化问题的有效方法，并使用核心集范式将其应用于现实世界的问题。作为该提案的一部分，核心集在大量无组织点云的曲线和表面重建等问题上的应用也将被探索。该提案的一个关键特征是，它将有助于推进跨学科的知识，如计算几何、凸优化和机器学习。核心集的使用将导致算法的设计更容易编码，更快，更健壮。此外，PI计划继续将他的研究成果纳入：(1)课程开发；(2)通过在理论、应用、算法和计算项目之间保持谨慎的平衡来指导研究生和本科生；(3)开发和维护与核心集及其应用相关的软件解决方案。

In a remarkably short time since their discovery, core-sets have emerged as a powerful tool for the study of geometric optimization problems. The main idea behind the technique is to compute a small subset of the input, called the Core-Set, that `approximates' the input and allows the optimization problem to be solved on this concise representation. Core-sets are exciting because of their prospect in the development of powerful, practical and general methods that are applicable to a large class of problems from various applications including automotive and aircraft industries, space exploration, civil surveyors, the movie, computer gaming and medical industries, and developers of security and geographic information systems.This NSF Career proposal focuses on the design and implementation of efficient methods for geometric optimization problems and their applications to real-world problems using the paradigm of core-sets. As part of this proposal, application of core-sets to problems like curve and surface reconstruction from massive unorganized point clouds will also be explored. A key characteristic of the proposal is that it will help advance the knowledge across disciplines like computational geometry, convex optimization and machine learning. The use of core-sets will result in the design of algorithms that will be easier to code and will be faster and more robust. Furthermore, the PI plans to continue incorporating his research results in: (1) course development, (2) mentoring of graduate and undergraduate students by maintaining a careful balance between theory, applications, algorithms and computational projects, (3) development and maintenance of software solutions related to core-sets and their applications.