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

这个教师早期职业发展（CAREER）计划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。在过去几年中，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）模型是重要的，从原始数据中提取有用的知识，高维问题，大数据量和固有的不确定性提出了重大挑战的优化算法的设计。本研究的目的是通过调查这些挑战：（i）新的一阶方法的确定性CP收敛速度更快，需要很少的结构信息，不依赖于线搜索，基于水平的方法;（ii）随机一阶方法，以最佳的方式处理数据的不确定性，基于随机近似;（iii）新的随机化方案，用于解决超出一阶方法能力的某些具有挑战性的确定性CP问题;和（iv）随机一阶和零阶方法一般，不一定凸，随机程序。研究集中在这些主题的两个基本问题：（i）复杂性的研究，提供了保证算法性能;（ii）结构的开发，导致设计具有更强复杂性和上级实际性能的算法。如果成功，一组新的算法方案将推进非线性和随机优化的最新技术，在易处理的范围内带来许多实际相关的数据分析问题。示例应用包括用于更快和更准确的医学图像重建和分类的算法，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，以改善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式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.