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

该项目将开发高效的基于梯度的创新算法和理论，用于解决大规模非线性优化问题，包括有或没有施加约束的问题。该研究将包括Barizilai-Borwein型梯度方法的渐近收敛性研究、有界约束优化的活动集技术和高效预处理器、连续背包约束问题的子空间仿射缩放方法以及具有线性等式约束相位的一般非线性优化的活动集方法。所有为优化而开发的技术，以及用于预处理器和投影仪的稀疏矩阵技术将被纳入用于一般大规模非线性优化的活动集算法中。将开发基于这项研究的高质量软件。尽管重点是非线性优化，但该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计算科学的许多领域产生广泛的影响。该项目的具体应用包括医学成像中出现的正电子发射断层扫描（PET），（2）拓扑优化中的设计问题，其中两种材料混合以优化其电性能，（3）支持向量机（SVM），用于模式识别和数据挖掘中的分离和分类问题，（4）算法并行化中出现的图分区以及稀疏矩阵分解的填充减少排序，以及（5）蛋白质折叠，其中蛋白质结构是从由核磁共振波谱得出的原子间距离。为了最大限度地发挥影响，该项目开发的软件将被广泛使用。受资助的研究生将接受数学跨学科应用的培训。

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.