Smoothing Methods in Optimization

优化中的平滑方法

• 批准号: 1514559
• 负责人: James Burke
• 金额: $ 18.7万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1514559&HistoricalAwards=false
• 关键词: Smoothing; Methods; Optimization

This research project concerns mathematical optimization, a field that has experienced explosive growth of over the past thirty years due to its wide applicability in science, engineering, business, and medicine. Contributing factors include the advent of the internet, advances in computational power and computing architectures, the availability of very large data sets, as well as advances in science, engineering, communication, and business. These developments have created a fertile ground for the emergence of new applications and data acquisition modalities, in addition to new methods for data management, interpretation, and modeling. These are the driving forces behind big data and machine learning research. In addition, there is a greater urgency in many disciplines for addressing questions concerning design, efficiency, risk, and inference, as well as model selection, system identification, and error and uncertainty quantification. This research project aims to develop new methods in non-smooth optimization and to study the practical impact of these methods. The research will emphasize underlying optimization tools, including model development, the design of numerical solution procedures, and the assessment and quantification of model validity, sensitivity, robustness, and uncertainty.This project concerns the methods and theory associated with the use of smoothing techniques in optimization. In order to extract solutions having pre-specified properties, objective functions in modern optimization problems are often non-differentiable, with the non-differentiability being a key descriptive component. In addition, non-differentiability is present in an essential way when the problem is constrained. Problems possessing non-differentiability appear across a broad spectrum of applications. These include robust statistical modeling, regularization formulations to encode prior information, system identification, sparsity optimization, matrix completion, semi-definite programming, and any problem possessing constraints. In addition, many modern problems are very large scale. Hence, there is a focus on the development of fast optimization algorithms for large-scale applications in the presence of non-smooth/non-convex objectives. Smoothing methods are designed to approximate these non-smooth problems, with the goal of rapidly obtaining good approximate solutions. This project is devoted to the development and understanding of new and emerging smoothing methods for non-smooth optimization, to providing a mathematical foundation for these methods, and to studying the practical impact of these methods on a range of applications. Primary objectives include (1) developing a framework for convergence analysis, (2) extending results for convex problems to non-convex problems, (3) providing a calculus for smoothing techniques, (4) developing error bounds using duality theory, (5) continued development of the level set method for optimization, and (6) consideration of parametrized optimal value functions.

该研究项目涉及数学优化，该领域由于其在科学，工程，商业和医学方面的广泛适用性而在过去三十年中经历了爆炸性的增长。贡献因素包括互联网的出现，计算能力和计算体系结构的进步，非常大的数据集的可用性以及科学，工程，沟通和业务的进步。除了用于数据管理，解释和建模的新方法外，这些发展为新应用程序和数据采集方式的出现创造了肥沃的基础。这些是大数据和机器学习研究背后的动力。此外，许多学科的紧迫性，要解决有关设计，效率，风险和推理以及模型选择，系统识别以及错误和不确定性量化的问题。该研究项目旨在开发非平滑优化的新方法，并研究这些方法的实际影响。 这项研究将强调基本的优化工具，包括模型开发，数值解决方案程序的设计以及模型有效性，敏感性，鲁棒性和不确定性的评估和量化。此项目涉及与在优化中使用平滑技术相关的方法和理论。为了提取具有预先指定属性的解决方案，现代优化问题中的目标函数通常是不可差异的，而非差异性是一个关键的描述组件。另外，当问题受到限制时，以必不可少的方式存在非差异性。具有非差异性的问题出现在广泛的应用中。其中包括可靠的统计建模，正规化公式以编码先前的信息，系统识别，稀疏性优化，矩阵完成，半定义编程以及任何具有约束的问题。此外，许多现代问题非常大。 因此，在存在非平滑/非凸目标的情况下，针对大规模应用程序的快速优化算法的开发。平滑方法旨在近似这些非平滑问题，目的是快速获得良好的近似解决方案。该项目致力于开发和理解非平滑优化的新和新兴平滑方法，为这些方法提供数学基础，并研究这些方法对一系列应用的实际影响。主要目标包括（1）开发一个用于收敛分析的框架，（2）扩展凸问题的结果到非convex问题，（3）为平滑技术提供微积分，（4）使用偶性理论开发误差界，（5）持续开发优化级别的级别设置方法，以及（6）（6）考虑到优化最佳值函数的考虑。

项目成果

Inexact Sequential Quadratic Optimization with Penalty Parameter Updates within the QP Solver

QP 求解器内带有惩罚参数更新的不精确序列二次优化

• DOI: 10.1137/18m1176488
• 发表时间: 2020
• 期刊: SIAM Journal on Optimization
• 影响因子: 3.1
• 作者: Burke, James V.;Curtis, Frank E.;Wang, Hao;Wang, Jiashan
• 通讯作者: Wang, Jiashan

The subdifferential of measurable composite max integrands and smoothing approximation

可测复合最大被积函数的次微分与平滑近似

• DOI: 10.1007/s10107-019-01441-9
• 发表时间: 2020
• 期刊: Mathematical Programming
• 影响因子: 2.7
• 作者: James V. Burke;Xiaojun Chen;Hailin Sun
• 通讯作者: Hailin Sun

Foundations of Gauge and Perspective Duality

• DOI: 10.1137/17m1119020
• 发表时间: 2017-02
• 期刊: SIAM J. Optim.
• 作者: A. Aravkin;J. Burke;D. Drusvyatskiy;M. Friedlander;Kellie J. MacPhee

Generalized Kalman smoothing: Modeling and algorithms

• DOI: 10.1016/j.automatica.2017.08.011
• 发表时间: 2017-12-01
• 期刊: AUTOMATICA
• 影响因子: 6.4
• 作者: Aravkin, Aleksandr;Burke, James V.;Pillonetto, Gianluigi
• 通讯作者: Pillonetto, Gianluigi

Variational Properties of Matrix Functions via the Generalized Matrix-Fractional Function

通过广义矩阵分数函数的矩阵函数的变分性质

• DOI: 10.1137/18m1209660
• 发表时间: 2019
• 期刊: SIAM Journal on Optimization
• 影响因子: 3.1
• 作者: Burke, James V.;Gao, Yuan;Hoheisel, Tim
• 通讯作者: Hoheisel, Tim