CAREER: Structure, Complexity, and Conditioning in Nonsmooth Optimization

近年来，地震学、数据科学、信息技术和环境科学等各个高影响领域的大数据集出现了前所未有的增长。从这样的数据集中提取有用信息的任务通常会导致解决大规模的优化问题。这类问题的巨大规模给优化专家带来了巨大的挑战。研究人员的目标是在这样的环境下推进大规模优化的覆盖范围，在整个科学和工程领域都有重要的应用。由此产生的方法和算法创建了一种系统的方法来发现观测数据背后的趋势和现象，并导致了对未观测数据进行预测的良好机制。该项目牢牢地处于理论和计算之间的交界处。因此，数值实验、教学和发现的有效组合是该提案的核心。研究生参与该项目的工作。研究人员的策略依赖于三个相互关联的支柱：结构、计算复杂性和非光滑优化的条件性。通过严格的收敛速度来判断数值方法的效率是最佳的。然而，如果算法的收敛保证与任何算法在问题类别中所能具有的最佳可能保证相匹配，则算法的收敛保证变得更加有效。寻找这样的“最佳方法”是计算复杂性的基础。衡量问题的条件性--表明其难度--在这一主题中发挥着核心作用，并与根本问题的稳定性密切相关。此外，理论和算法还得益于应用中广泛存在的丰富的底层结构，如光滑和简单非光滑泛函分量的分离、光滑的共轭表示、鞍点重构等。凸优化技术和变分分析洞察力指导着研究者的方法。数据科学和工程中普遍存在的大规模问题可以直接受益于这项工作。该项目将各方面的研究和教学融为一体。研究生参与该项目的工作。

Recent years have seen an unprecedented growth of large data sets in various high-impact fields, such as seismology, data science, information technology, and environmental science. The task of extracting useful information from such data sets typically leads to solving a large-scale optimization problem. The sheer size of such problems poses great challenges for optimization specialists. The investigator aims to advance the reach of large-scale optimization in such settings, with vital applications throughout science and engineering. The resulting methodology and algorithms create a systematic approach to discover trends and phenomena underlying the observed data, and lead to a well-grounded mechanism for making predictions about unobserved data. The project lies firmly at the interface between theory and computation. Therefore, an effective mix of numerical experimentation, teaching, and discovery is central to the proposal. Graduate students participate in the work of the project. The investigator's strategy rests on three interrelated pillars: structure, computational complexity, and conditioning in nonsmooth optimization. Efficiency of numerical methods is best judged through rigorous rates of convergence. Convergence guarantees of an algorithm become much more potent, however, if they match best possible guarantees that any algorithm can have within the problem class. The search for such "optimal methods" underpins computational complexity. Measures of the problem's conditioning -- an indication of its difficulty -- play a central role in the subject and are intimately tied to stability of the underlying problem. The theory and algorithms, moreover, benefit greatly from exploiting rich underlying structure prevalent in applications, such as separation of smooth and simple nonsmooth functional components, smooth conjugate representations, saddle-point reformulations, etc. Convex optimization techniques and variational analytic insight guide the investigator's approach. Pervasive large-scale problems in data science and engineering can directly benefit from this work. The project integrates research and teaching in all aspects. Graduate students participate in the work of the project.