Collaborative Research: AF: Medium: Fundamental Challenges in Optimization

有效的优化是现代计算的基础。 在过去的世纪的进步导致了强大的技术来解决问题，从无数的应用。 求解线性系统、线性规划和整数规划等基本问题的复杂性是算法理论的核心。 该项目的目标是通过解决有效优化前沿的挑战性问题来推进该领域。 新的算法所考虑的问题将直接感兴趣的许多学科在科学和工程。 该项目包括一个针对初级研究人员的研究讲习班，以及一个由研究人员团队提供的关于现代优化技术的视频辅导课程。 该项目的研究成果正在被纳入“连续算法”、“谱算法”和“整数和线性规划”课程中。 该项目产生的课程内容和软件正在公开提供。该项目的重点问题从连续到离散不等：（1）稀疏一般线性系统，（2）稀疏线性规划和p-范数最小化，（3）使用半定规划来有效地低秩逼近组合优化问题，（4）提出了离散优化的连续化方法;（5）解决了整数规划的复杂性问题。 离散优化的最新进展在很大程度上依赖于连续方法的启发和分析，而连续优化问题的解决通常使用图论。 预计将开发出弥合这一范围的、具有广泛普遍适用性的新技术。与运筹学、数值分析、随机过程（包括各种类型的扩散）、同伦方法、图论、凸性的几何性质和格理论的联系正在得到加强。该奖项反映了NSF的法定使命，并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。

Efficient Optimization is the bedrock of modern computation. Progress over the last century has led to powerful techniques to solve problems from myriad applications. The complexity of basic questions such as solving linear systems, linear programs and integer programs is at the core of the theory of algorithms. The goal of this project is to advance the field by addressing challenging problems on the frontier of efficient optimization. New algorithms for the problems considered would be of direct interest to many disciplines in science and engineering. The project includes a research workshop targeted at junior researchers, as well as a video tutorial course by the team of investigators on contemporary techniques in optimization. Research findings from the project are being incorporated into courses on “Continuous Algorithms”, “Spectral Algorithms” and “Integer and Linear Programming”. Course content and software resulting from the project is being made publicly available.The focus problems of this project range from continuous to discrete: (1) sparse general Linear Systems, (2) sparse Linear Programs and p-norm minimization, (3) using Semi-definite Programs for efficient low-rank approximation to combinatorial optimization problems, (4) advancing the continuous approach to discrete optimization and (5) resolving the complexity of integer programming. Recent progress on discrete optimization has relied heavily on inspiration and analysis from continuous methods, while the solution of continuous optimization problems often uses graph theory. It is anticipated that new techniques bridging this spectrum, and of broad, general applicability, will be developed. The connections to operations research, numerical analysis, stochastic processes (including various types of diffusion), homotopy methods, graph theory, geometric properties of convexity and the theory of lattices are being enhanced.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.