Novel Numerical Approaches for Structured Optimization

许多现代应用程序中涉及的数据集非常大，并且通常在分布式位置收集，并且随着时间的推移而不断进行。常见的例子是与互联网、社交网络、信息技术、医疗保健、生物学、金融和工程上的搜索相关联的数据集。分析和学习这些海量数据集对计算、数据存储和数据传输提出了巨大的挑战。另一方面，高性能计算机现在已经很容易获得。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。 本项目以本科生和研究生为对象，旨在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。定义的目标函数和最优解都表现出特定的结构，包括凸性，光滑性和多线性的前者，稀疏性，低秩，和正交的后者。本研究旨在利用这种结构设计有效的计算方法。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度分析。随机近似将用于处理涉及流数据的问题，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调查。该研究的目的是显着加快现有的算法在理论上和实践中，导致现有的算法，目前缺乏收敛性分析的新的理论结果，并产生新的算法计算解决方案的复杂问题，目前不能有效地解决。

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.