Smoothing Methods in Optimization

University Of Washington, Seattle WA

Investigators

Abstract

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.