The Analysis and Design of Gradient Methods for Large-Scale Nonlinear Optimization and Applications

大规模非线性优化的梯度法分析与设计及应用

• 批准号: 1016204
• 负责人: Hongchao Zhang
• 金额: $ 14.16万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016204&HistoricalAwards=false
• 关键词: Analysis; Design; Gradient; Methods; Large

This project will develop efficient gradient-based innovative algorithms and theory for the solution of large-scale nonlinear optimization problems, including those with and without constraints imposed. The research will include the asymptotic convergence studies of Barizilai-Borwein type gradient methods, active set techniques and efficient preconditioners for bound constrained optimization, subspace affine-scaling methods for problems with continuous knapsack constraints, and active set methods for general nonlinear optimization with linear equality constrained phase. All the techniques developed for the optimization, as well as the sparse matrix technology for preconditioners and projectors will be incorporated into the active set algorithm for general large-scale nonlinear optimization. High quality software based on this research will be developed.Although the focus is nonlinear optimization, the methods and algorithms developed in the project will have broad impact in the many areas of computational science that require the solution of such large-scale nonlinear optimization problems. Specific applications of this project include Positron Emission Tomography (PET) which arises in medical imaging, (2) a design problem in topology optimization where two materials are mixed so as to optimize their electrical properties, (3) Support Vector Machines (SVM) which are used in separation and classification problems in pattern recognition and data mining, (4) graph partitioning which arises in parallelization of algorithms and in fill reducing orderings for sparse matrix factorization, and (5) protein folding where the structure of a protein is reconstructed from inter-atomic distances derived from nuclear magnetic resonance spectroscopy. To maximize the impact,the software developed in this project will be made widely available. The supported graduate student will receive training on interdisciplinary applications of mathematics.

该项目将开发有效的基于梯度的创新算法和理论，以解决大规模的非线性优化问题，包括施加有限的那些。这项研究将包括Barizilai-Borwein型梯度方法的渐近收敛研究，主动集合技术和有效的预处理，用于约束优化的约束优化，子空间仿射尺度方法，用于连续背包约束的问题，以及具有一般非线性优化的一般非线性限制性相位的活性固定方法。为优化开发的所有技术，以及用于预处理器和投影仪的稀疏矩阵技术，将纳入一般大规模非线性优化的活动集合算法中。基于这项研究的高质量软件将开发。尽管重点是非线性优化，但项目中开发的方法和算法将在需要解决此类大规模非线性优化问题的许多计算科学领域产生广泛的影响。该项目的特定应用包括在医学成像中产生的正电子发射断层扫描（PET），（2）拓扑优化中的设计问题，其中两种材料混合在一起以优化其电气特性，（3）支持向量机器（SVM），用于分离和分类用于模式识别和数据挖掘的分类问题，（4）在填充平面化的阶段，并在平面相平面化，并在平面化中填充平面化，并将其用于平面化。分解和（5）蛋白质折叠，其中蛋白质的结构是从核磁共振光谱衍生的原子间距离重建的。为了最大程度地发挥影响力，将在该项目中开发的软件可广泛使用。受支持的研究生将接受有关数学跨学科应用的培训。