# Direct sparse matrix methods for interior point algorithms.

http://hdl.handle.net/10150/185133
Title:
Direct sparse matrix methods for interior point algorithms.
Author:
Jung, Ho-Won.
Issue Date:
1990
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
Recent advances in linear programming solution methodology have focused on interior point algorithms. These are powerful new methods, achieving significant reductions in computer time for large LPs and solving problems significantly larger than previously possible. This dissertation describes the implementation of interior point algorithms. It focuses on applications of direct sparse matrix methods to sparse symmetric positive definite systems of linear equations on scalar computers and vector supercomputers. The most computationally intensive step in each iteration of any interior point algorithm is the numerical factorization of a sparse symmetric positive definite matrix. In large problems or relatively dense problems, 80-90% or more of computational time is spent in this step. This study concentrates on solution methods for such linear systems. It is based on modifications and extensions of graph theory applied to sparse matrices. The row and column permutation of a sparse symmetric positive definite matrix dramatically affects the performance of solution algorithms. Various reordering methods are considered to find the best ordering for various numerical factorization methods and computer architectures. It is assumed that the reordering method will follow the fill-preserving rule, i.e., not allow additional fill-ins beyond that provided by the initial ordering. To follow this rule, a modular approach is used. In this approach, the matrix is first permuted by using any minimum degree heuristic, and then the permuted matrix is again reordered according to a specific reordering objective. Results of different reordering methods are described. There are several ways to compute the Cholesky factor of a symmetric positive definite matrix. A column Cholesky algorithm is a popular method for dense and sparse matrix factorization on serial and parallel computers. Applying this algorithm to a sparse matrix requires the use of sparse vector operations. Graph theory is applied to reduce sparse vector computations. A second and relatively new algorithm is the multifrontal algorithm. This method uses dense operations for sparse matrix computation at the expense of some data manipulation. The performance of the column Cholesky and multifrontal algorithms in the numerical factorization of a sparse symmetric positive definite matrix on an IBM 3090 vector supercomputer is described.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Sparse matrices; Trees (Graph theory); Linear programming.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Degree Grantor:
University of Arizona
Marsten, Roy E.

DC FieldValue Language
dc.language.isoenen_US
dc.titleDirect sparse matrix methods for interior point algorithms.en_US
dc.creatorJung, Ho-Won.en_US
dc.contributor.authorJung, Ho-Won.en_US
dc.date.issued1990en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractRecent advances in linear programming solution methodology have focused on interior point algorithms. These are powerful new methods, achieving significant reductions in computer time for large LPs and solving problems significantly larger than previously possible. This dissertation describes the implementation of interior point algorithms. It focuses on applications of direct sparse matrix methods to sparse symmetric positive definite systems of linear equations on scalar computers and vector supercomputers. The most computationally intensive step in each iteration of any interior point algorithm is the numerical factorization of a sparse symmetric positive definite matrix. In large problems or relatively dense problems, 80-90% or more of computational time is spent in this step. This study concentrates on solution methods for such linear systems. It is based on modifications and extensions of graph theory applied to sparse matrices. The row and column permutation of a sparse symmetric positive definite matrix dramatically affects the performance of solution algorithms. Various reordering methods are considered to find the best ordering for various numerical factorization methods and computer architectures. It is assumed that the reordering method will follow the fill-preserving rule, i.e., not allow additional fill-ins beyond that provided by the initial ordering. To follow this rule, a modular approach is used. In this approach, the matrix is first permuted by using any minimum degree heuristic, and then the permuted matrix is again reordered according to a specific reordering objective. Results of different reordering methods are described. There are several ways to compute the Cholesky factor of a symmetric positive definite matrix. A column Cholesky algorithm is a popular method for dense and sparse matrix factorization on serial and parallel computers. Applying this algorithm to a sparse matrix requires the use of sparse vector operations. Graph theory is applied to reduce sparse vector computations. A second and relatively new algorithm is the multifrontal algorithm. This method uses dense operations for sparse matrix computation at the expense of some data manipulation. The performance of the column Cholesky and multifrontal algorithms in the numerical factorization of a sparse symmetric positive definite matrix on an IBM 3090 vector supercomputer is described.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectSparse matricesen_US
dc.subjectTrees (Graph theory)en_US
dc.subjectLinear programming.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.committeememberSaltzman, Matthew J.en_US
dc.contributor.committeememberKannan, Pallassanaen_US
dc.contributor.committeememberSheng, Olivia R.en_US
dc.contributor.committeememberGoldberg, Jeffrey B.en_US
dc.identifier.proquest9100551en_US
dc.identifier.oclc704729031en_US