Smoothing Methods in Optimization

本研究项目涉及数学优化，由于其在科学、工程、商业和医学领域的广泛应用，在过去三十年中经历了爆炸式增长。促成因素包括互联网的出现、计算能力和计算架构的进步、超大数据集的可用性，以及科学、工程、通信和商业的进步。除了数据管理、解释和建模的新方法之外，这些发展还为新应用程序和数据采集模式的出现创造了肥沃的土壤。这些都是大数据和机器学习研究背后的驱动力。此外，在许多学科中，对于解决有关设计、效率、风险和推理、以及模型选择、系统识别、错误和不确定性量化的问题，存在更大的紧迫性。本课题旨在开发非光滑优化的新方法，并研究这些方法的实际影响。该研究将强调潜在的优化工具，包括模型开发，数值解程序的设计，以及模型有效性，灵敏度，鲁棒性和不确定性的评估和量化。这个项目涉及的方法和理论与使用平滑技术的优化。为了提取具有预定性质的解，现代优化问题中的目标函数通常是不可微的，不可微性是一个关键的描述成分。此外，当问题有约束时，不可微性以一种必要的方式存在。具有不可微性的问题出现在广泛的应用中。这些包括稳健的统计建模、用于编码先验信息的正则化公式、系统识别、稀疏性优化、矩阵补全、半确定规划和任何具有约束的问题。此外，许多现代问题都是非常大规模的。因此，在存在非光滑/非凸目标的大规模应用中，有一个重点是开发快速优化算法。设计光滑方法来近似这些非光滑问题，目的是快速获得良好的近似解。本项目致力于发展和理解用于非光滑优化的新兴平滑方法，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)建立一个收敛分析的框架，(2)将凸问题的结果扩展到非凸问题，(3)为平滑技术提供微积分，(4)使用对偶理论开发误差界限，(5)继续发展优化的水平集方法，以及(6)考虑参数化最优值函数。

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.