Regularized Symmetric Indefinite Systems in Interior Point Methods for Linear and Quadratic Optimization

This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraint. The new regularization techniques for Newton equation system applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidefinite systems known to be strongly factorizable to a form of Cholesky-like factorization.

[1]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[2]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[3]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..

[4]  M. Kojima,et al.  A primal-dual interior point algorithm for linear programming , 1988 .

[5]  N. Megiddo.  Pathways to the optimal set in linear programming , 1989 .

[6]  I. Duff,et al.  On the augmented system approach to sparse least-squares problems , 1989 .

[7]  R. Setiono.  Interior proximal point algorithm for linear programs , 1992 .

[8]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

[9]  Roy E. Marsten,et al.  On Implementing Mehrotra's Predictor-Corrector Interior-Point Method for Linear Programming , 1992, SIAM J. Optim..

[10]  Sanjay Mehrotra,et al.  Solving symmetric indefinite systems in an interior-point method for linear programming , 1993, Math. Program..

[11]  John M. Mulvey,et al.  Higher-Order Predictor-Corrector Interior Point Methods with Application to Quadratic Objectives , 1993, SIAM J. Optim..

[12]  Robert J. Vanderbei,et al.  Symmetric indefinite systems for interior point methods , 1993, Math. Program..

[13]  J. Gondzio.  Implimenting cholesky factorization for interior point methods of linear programming , 1993 .

[14]  Roy E. Marsten,et al.  Feature Article - Interior Point Methods for Linear Programming: Computational State of the Art , 1994, INFORMS J. Comput..

[15]  J. Gondzio.  HOPDM (version 2.12) — A fast LP solver based on a primal-dual interior point method , 1995 .

[16]  Robert J. Vanderbei,et al.  Symmetric Quasidefinite Matrices , 1995, SIAM J. Optim..

[17]  Anna Altman.  QHOPDM -- A higher order primal-dual method for large scale convex quadratic programming , 1995 .

[18]  Jacek Gondzio,et al.  Multiple centrality corrections in a primal-dual method for linear programming , 1996, Comput. Optim. Appl..

[19]  Jacek Gondzio,et al.  Implementation of Interior Point Methods for Large Scale Linear Programming , 1996 .

[20]  Gisela Engeln-Müllges,et al.  Direct Methods for Solving Systems of Linear Equations , 1996 .

[21]  Markus Amann,et al.  Cost-effective Sulphur Emission Reduction under Uncertainty , 1996 .

[22]  M. Saunders,et al.  SOLVING REGULARIZED LINEAR PROGRAMS USING BARRIER METHODS AND KKT SYSTEMS , 1996 .

[23]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[24]  C. Mészáros,et al.  A repository of convex quadratic programming problems , 1999 .

[25]  R. Vanderbei.  LOQO:an interior point code for quadratic programming , 1999 .