Novel Numerical Approaches for Structured Optimization

在许多现代应用中涉及的数据集都是非常大的，并且通常是在分布的位置收集的，并且随着时间的推移是连续的。常见的例子是与互联网、社交网络、信息技术、医疗保健、生物、金融和工程上的搜索相关的数据集。分析和学习这些海量的数据集给计算、数据存储和数据传输带来了巨大的挑战。另一方面，高性能计算机现在唾手可得。该项目旨在开发新的计算方法，以实现涉及超大数据集的问题的高性能计算。这些方法解决了数据科学和工程应用中出现的几个计算挑战。本科生和研究生都参与了这个项目。该项目致力于设计新颖的计算算法并分析其理论行为，以解决涉及大量数据集并由大量变量参数化的结构化优化问题。定义目标函数和最优解都表现出特定的结构，包括前者的凸性、光滑性和多线性性，后者的稀疏性、低秩性和正交性。本研究旨在利用这种结构设计高效的计算方法。该项目包括几个研究方向，从处理复杂正则化的变量分裂，到自适应异步并行计算和收敛速率分析。随机近似将用于处理涉及流数据的问题，而新的数值方法将用于通过原始对偶更新来解决非线性约束问题。多阵列结构的问题也将被研究。研究的目的是在理论和实践上显著提高现有算法的速度，对现有算法缺乏收敛性分析的问题提出新的理论结果，并对目前无法有效求解的复杂问题提出新的计算算法。

The data sets involved in many modern applications are extremely large, and are often collected at distributed locations and continuously with the progression of time. Common examples are data sets associated with searches on the Internet, social networks, information technology, healthcare, biology, finance, and engineering. Analyzing and learning from these massive data sets imposes great challenges on computation, data storage, and data transfer. On the other hand, high performance computers are now readily available. This project aims to develop novel computational methods to enable high performance computing for questions involving extremely large data sets. The approaches address several computational challenges that emerge from applications across data sciences and engineering. Undergraduate and graduate students are involved in the project.This project is focused on designing novel computational algorithms and analyzing their theoretical behaviors for solving structured optimization problems that involve huge data sets and are parameterized by large numbers of variables. Both the defining objective functions and the optimal solutions exhibit particular structures, including convexity, smoothness, and multi-linearity for the former, and sparsity, low-rank, and orthogonality for the latter. This research aims to take advantage of this structure in designing efficient computational methods. The project includes several research directions, from variable splitting for handling complicated regularizers, to adaptive asynchronous parallel computing and analysis of convergence rates. Stochastic approximations will be used for dealing with problems involving stream data, and novel numerical approaches will be used to solve non-linearly constrained problems via primal-dual updates. Problems with multi-array structure will also be investigated. The research aims to significantly speed up existing algorithms both theoretically and practically, lead to new theoretical results of existing algorithms that currently lack convergence analysis, and give rise to novel algorithms for computing solutions to complicated problems that are currently not efficiently solvable.