Ordering Heuristics in Interior Point LP Methods
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.
KeywordsMinimum Degree Interior Point Method Cholesky Factor Sparsity Pattern True Degree
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