## Stability of Augmented System Factorizations in Interior-Point Methods (1997)

Venue: | SIAM J. Matrix Anal. Appl |

Citations: | 16 - 2 self |

### BibTeX

@ARTICLE{Wright97stabilityof,

author = {Stephen Wright},

title = {Stability of Augmented System Factorizations in Interior-Point Methods},

journal = {SIAM J. Matrix Anal. Appl},

year = {1997},

volume = {18},

pages = {191--222}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Some implementations of interior-point algorithms obtain their search directions by solving symmetric indefinite systems of linear equations. The conditioning of the coefficient matrices in these so-called augmentedsystems deteriorates on later iterations, as some of the diagonal elements grow without bound. Despite this apparent difficulty, the steps produced by standard factorization procedures are often accurate enough to allow the interior-point method to converge to high accuracy. When the underlying linear program is nondegenerate, we show that convergence to arbitrarily high accuracy occurs, at a rate that closely approximates the theory. We also explain and demonstrate what happens when the linear program is degenerate, where convergence to acceptable accuracy (but not arbitrarily high accuracy) is usually obtained. 1. Introduction. We focus on the core linear algebra operation in primal-dual interior-point methods for linear programming: solution of a system of linear equat...

### Citations

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Citation Context ...ype described in this paper. They compute the maximum step ff that could be taken along this direction without violating the positivity bounds, then set the actual step length to .995 ff . Mehrotra's =-=[14]-=- predictor-corrector search direction differs from the one analyzed in this paper, but under our assumptions below, the difference vanishes as the solution is approached. Newer codes, such as those de... |

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Citation Context ...lace in the context of linear complementarity, a class of problems that includes linear programming as a special case. On the computational side, the OB1 code described by Lustig, Marsten, and Shanno =-=[10]-=- generated search directions of the type described in this paper. They compute the maximum step ff that could be taken along this direction without violating the positivity bounds, then set the actual... |

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Citation Context ...cessful interior-point methods for practical linear programming problems are primal-dual methods. The best-known potential-reduction algorithm in this class was devised by Kojima, Mizuno, and Yoshise =-=[9]-=-; the review paper of Todd [17] contains a wealth of historical information on potential-reduction methods. Early developments in path-following methods are surveyed by Gonzaga [7], while Mizuno, Todd... |

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Citation Context ...fference vanishes as the solution is approached. Newer codes, such as those described by Mehrotra [14], Fourer and Mehrotra [5], Lustig, Marsten, and Shanno [12], Vanderbei [18], and Xu, Hung, and Ye =-=[23]-=- all implement Mehrotra's predictor-corrector strategy. These newer codes continue to use step lengths based on ff ; hence, we pay particular attention to the effect of roundoff error on this quantity... |

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Citation Context ...--- the BunchParlett, Bunch-Kaufman, and sparse Bunch-Parlett algorithms. The last of these has been used in at least one practical interior-point code for linear programming (see Fourer and Mehrotra =-=[5]-=-). We assume that no attempt is made to improve the conditioning of the underlying linear systems by guessing whether each component of the solution is at a bound. Preprocessing of this kind detracts ... |

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Citation Context ...he results overlap. However, they assume that the factorization algorithms select the large diagonal elements as pivots before any others, a pattern that does not generally occur in practice. Vavasis =-=[19]-=- gives an illuminating discussion of the augmented system in contexts other than optimization. He presents a solution method that is provably stable in a certain sense, but which is not guaranteed to ... |

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LOQO user’s manual
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Citation Context ...r assumptions below, the difference vanishes as the solution is approached. Newer codes, such as those described by Mehrotra [14], Fourer and Mehrotra [5], Lustig, Marsten, and Shanno [12], Vanderbei =-=[18]-=-, and Xu, Hung, and Ye [23] all implement Mehrotra's predictor-corrector strategy. These newer codes continue to use step lengths based on ff ; hence, we pay particular attention to the effect of roun... |

19 |
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Citation Context ...an elimination in the context of interior-point algorithms for linear complementarity problems. Simultaneously with the original version of this paper, and independently, Forsgren, Gill, and Shinnerl =-=[4]-=- performed an analysis of the augmented system in barrier algorithms. Their analysis tends to be more detailed than ours, and a few of the results overlap. However, they assume that the factorization ... |

13 |
On the Convergence of a Class of InfeasibleInterior-Point Methods for the Horizontal Linear Complementarity Problem
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(Show Context)
Citation Context ...eyed by Gonzaga [7], while Mizuno, Todd, and Ye [15] describe an important variant of these methods that does not require the iterates to stay within a cramped neighborhood of the central path. Zhang =-=[25]-=- extended the path-following approach further, allowing the iterates to be infeasible while retaining global convergence and polynomial complexity; see also Wright [21]. Some of these developments too... |

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Citation Context ...m in contexts other than optimization. He presents a solution method that is provably stable in a certain sense, but which is not guaranteed to produce "useful" steps in the sense of this pa=-=per. Duff [3]-=- also discusses augmented systems in a general context and describes a sparse factorization procedure. 2. Interior-Point Methods. We consider the linear program in standard form: min c T x; Ax = b; xs... |

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Citation Context ... for practical linear programming problems are primal-dual methods. The best-known potential-reduction algorithm in this class was devised by Kojima, Mizuno, and Yoshise [9]; the review paper of Todd =-=[17]-=- contains a wealth of historical information on potential-reduction methods. Early developments in path-following methods are surveyed by Gonzaga [7], while Mizuno, Todd, and Ye [15] describe an impor... |

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Citation Context ...attention to the effect of roundoff error on this quantity. Previous analysis of the ill-conditioned linear systems that arise in interior-point and barrier methods has been carried out by Poncele'on =-=[16]-=- and Wright [22]. Poncele 'on [16] showed that these systems are not too sensitive to structured perturbations from a certain class provided that the underlying optimization problem is well conditione... |

2 |
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(Show Context)
Citation Context ... effect of roundoff error on this quantity. Previous analysis of the ill-conditioned linear systems that arise in interior-point and barrier methods has been carried out by Poncele'on [16] and Wright =-=[22]-=-. Poncele 'on [16] showed that these systems are not too sensitive to structured perturbations from a certain class provided that the underlying optimization problem is well conditioned. Wright [22] a... |