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

Louisiana State University, Baton Rouge LA

Investigators

Abstract

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.