CAREER: Reduced-order Methods for Big-Data Challenges in Nonlinear and Stochastic Optimization

这个教师早期职业发展（Career）项目的目标是开发一套新的降阶算法，以解决优化中的大数据挑战。过去几年，可用数据的数量出现了前所未有的增长。虽然非线性，特别是凸规划（CP）模型对于从原始数据中提取有用的知识非常重要，但高问题维数、大数据量和固有的不确定性对优化算法的设计提出了重大挑战。本研究旨在通过研究：(i)基于水平方法的一阶确定性CP新方法，该方法收敛速度更快，需要很少的结构信息并且不依赖于线搜索；（ii）基于随机近似，以最优方式处理数据不确定性的随机一阶方法；（iii）解决一阶方法无法解决的某些具有挑战性的确定性CP问题的新颖随机化方案；（iv）一般（不一定是凸的）随机规划的随机一阶和零阶方法。研究的重点是跨越这些主题的两个基本问题：(i)复杂性的研究，它提供了算法性能的保证；（ii）利用结构设计出复杂度更强、实用性能更优的算法。如果成功，一套新的算法方案将推动非线性和随机优化的最新发展，使许多实际相关的数据分析问题在可追溯的范围内。示例应用包括更快和更准确的医学图像重建和分类算法，这将有利于医疗保健。此外，在地震学中，有效的随机规划方法将有助于通过测量地震台站检测到的数千次地震来建立预测模型。该项目还将支持PI的教育目标，即提高学生在运筹学方面的学习，扩大博士课程中代表性不足的群体的代表性，并通过开发优化求解器为开放研究基础设施做出贡献。

The objective of this Faculty Early Career Development (CAREER) Program project is to develop a set of new reduced-order algorithms to tackle the big-data challenges in optimization. The last several years have seen an unprecedented growth in the amount of available data. While nonlinear, especially convex programming (CP) models are important to extract useful knowledge from raw data, high problem dimensionality, large data volumes and inherent uncertainty present significant challenges to the design of optimization algorithms. This research aims to attack these challenges by investigating: (i) novel first-order methods for deterministic CP that converge faster, require little structural information and do not rely on line search, based on level methods; (ii) stochastic first-order methods that handle data uncertainty in an optimal manner, based on stochastic approximation; (iii) novel randomization schemes for solving certain challenging deterministic CP problems beyond the capability of first-order methods; and (iv) stochastic first- and zeroth-order methods for general, not necessarily convex, stochastic programs. The research focuses on two fundamental issues across these topics: (i) the study of complexity which provides guarantees on algorithmic performance; and (ii) the exploitation of structures that leads to the design of algorithms with stronger complexity and superior practical performance.If successful, a set of new algorithmic schemes will advance the state-of-the-art in nonlinear and stochastic optimization, bringing many practically relevant data analysis problems within the range of tractability. Example applications include algorithms for faster and more accurate medical image reconstruction and classification, which will be beneficial to healthcare. In addition, in seismology, effective stochastic programming methods will help to build predictive models by measuring thousands of earthquakes detected at seismic stations. The project will also support the PI's educational goals to improve students' learning in operations research, broaden the representation of underrepresented groups in the PhD program, and contribute to open research infrastructure through the development of optimization solvers.