New Trends in Mathematical Programming pp 203-221 | Cite as

# Ordering Heuristics in Interior Point LP Methods

## Abstract

The success of the implementation of the normal equations approach of interior point methods (IPM) for linear programming depends on the quality of its *analysis* phase, i.e. reordering for sparsity. The goal of this analysis is to find a permutation matrix *P* such that the Cholesky factor of *PAD* ^{2} *A* ^{ T } *P* ^{ T } is the sparsest possible. In practice, heuristics are used to solve this problem because finding an optimal permutation is an NP-complete problem. Two such heuristics, namely the *minimum degree* and the *minimum local fill—in* orderings are particularly useful in the context of IPM implementations. In this paper a parametric set of symbolic orderings is presented, which connects these two major approaches. It will be shown that in the “neighborhood” of the minimum degree ordering a practically efficient method exist. Implementation details will be discussed as well, and on a demonstrative set of linear programming test problems the performance of the new method will be compared with Sparspak’s GENQMD subroutine which was for a long time public accesible from NETLIB, and with the minimum local fill—in ordering implementation of CPLEX version 4.0.

## Keywords

Minimum Degree Interior Point Method Cholesky Factor Sparsity Pattern True Degree## Preview

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