Theory and Efficient Algorithms for Hard, Large Scale, Numerical Optimization

大规模硬数值优化的理论和高效算法

• 批准号: 9161-2013
• 负责人: Wolkowicz, Henry
• 金额: $ 2.48万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568649
• 关键词: Theory; Efficient; Algorithms; Hard; Large

The focus of my research will be the proper modelling of hard problems, and the design and implementation of efficient and robust numerical algorithms for large scale, hard, optimization problems. The problems I will deal with arise in many important applications, e.g. molecular conformation (MC), sensor network localization (SNL), inverse imaging and machine learning. In particular, many of these problems arise in the relaxations of hard combinatorial optimization problems. In many instances, the usual modelling approaches result in problems that are both large scale and ill-posed. Therefore, they are hard to solve numerically. Rather than being a disadvantage, one can often take advantage of the ill-posedness to get both a stable problem and one that is smaller in size. In particular, for problems such as SNL one can solve huge problems to high accuracy by exploiting the hidden degeneracy. I plan on applying this technique to MC problems with noisy data as well as to protein design problems. The techniques that I use involve continuous optimization, nonlinear programming (NLP) and in particular, semidefinite programming (SDP). For SDP, my research contributions involve theory, algorithms, and applications, i.e., they include strong duality results, stable algorithms, and the study of relaxations for various applications. Some of the codes that I have designed for the standard form Linear Programming (LP) model have already been implemented in MAPLE. I plan to implement algorithms for more general LP models that include both upper and lower bounds on the variables. As well I plan on implementing codes that solve more general NLP problems. For the NLP problems, I use algorithms for generalized trust region problems to solve large scale unconstrained minimization, as well as solve general NLP using stable sequential quadratic programming methods. The success of these implementations means that researchers in my field will have access to stable, high accuracy, algorithms.

我的研究重点将是困难问题的适当建模，以及为大规模，困难的优化问题设计和实现高效和鲁棒的数值算法。我将处理的问题出现在许多重要的应用中，例如分子构象（MC），传感器网络定位（SNL），逆成像和机器学习。特别是，许多这类问题出现在困难组合优化问题的松弛中。在许多情况下，通常的建模方法会导致大规模和病态的问题。因此，它们很难用数值方法求解。一个人常常可以利用身体不适来得到一个稳定的问题和一个较小的问题，而不是一个劣势。特别是对于SNL这样的问题，人们可以利用隐藏的简并性来高精度地解决巨大的问题。我计划将这种技术应用于带有噪声数据的MC问题以及蛋白质设计问题。