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ABSTRACT: The Netlib library of linear programming problems is a well known suite containing many real world applications. Recently it was shown by Ordonez and Freund that 71% of these problems are ill-conditioned. Hence, numerical difficulties may occur. Here, we present rigorous results for this library that are computed by a verification method using interval arithmetic. In addition to the original input data of these problems we also consider interval input data. The computed rigorous bounds and the performance of the algorithms are related to the distance to the next ill-posed linear programming problem.
Reliable Computing, Volume 12, Issue 4, Pages 303 - 321.
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ABSTRACT: A wide variety of problems in global optimization, combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver. It turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver. Numerical results are presented including problems from the SDPLIB and the NETLIB LP library; these libraries contain many ill-conditioned and real life problems.
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Christian Keil
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ABSTRACT: This dissertation treats the theory, implementation, and application of rigorous error bounds in the context of finite dimensional linear programming problems. Despite the theory of linear programming that is well understood and its numerous applications, commercial solvers frequently produce erroneous results for these problems. In contrast, verification methods yield solutions proved to be correct. The thesis presents theorems that yield rigorous error bounds, a convergence analysis, and generalizations. The software package Lurupa is described, which offers the rigorous error bounds as a standalone software, a library, and from MATLAB. Extensive numerical experiments and a comparison with other software packages are presented. They demonstrate that exploiting the special structure of a problem is necessary when aiming for fast and reliable results. Die Dissertation behandelt die Theorie, Implementierung und Anwendung rigoroser Fehlerschranken für endlich-dimensionale lineare Programme. Trotz der gut verstandenen Theorie und zahlreicher Anwendungen liefern kommerzielle Softwarepakete häufig falsche Resultate für diese Probleme. Im Gegensatz dazu liefern Verifikationsmethoden nachweislich korrekte Ergebnisse. Die Arbeit präsentiert Theoreme, die solche Schranken liefern, begleitet von einer Konvergenzanalyse und Verallgemeinerungen. Das Softwarepaket Lurupa wird beschrieben, welches die rigorosen Schranken als eigenständiges Paket, als Bibliothek und von MATLAB aus zur Verfügung stellt. Ausführliche numerische Experimente und ein Vergleich mit anderen Softwarepaketen werden präsentiert. Diese demonstrieren die Notwendigkeit die spezielle Struktur eines Problems auszunutzen, um schnelle und zuverlässige Ergebnisse zu erhalten.
Christian Keil: Rigorous Error Bounds for Finite Dimensional Linear Programming Problems. Norderstedt: Books on Demand GmbH, 2009.
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Acta Forestalia Fennica. 1968. 82(4): 1-35.
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Christian Keil
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ABSTRACT: Linear Programming has numerous applications, e.g., operations research, relaxations in global optimization, computational geometry. Recently it has been shown that many real world problems exhibit numerical difficulties due to ill-conditioning. Lurupa is a software package for computing rigorous optimal value bounds. The package can handle point and interval problems. Numerical experience with the Netlib lp library is given. @InProceedings{keil:DSP:2006:445, author = {Christian Keil}, title = {Lurupa - Rigorous Error Bounds in Linear Programming}, booktitle = {Algebraic and Numerical Algorithms and Computer-assisted Proofs}, year = {2006}, editor = {Bruno Buchberger and Shin'ichi Oishi and Michael Plum and Sigfried M. Rump}, number = {05391}, series = {Dagstuhl Seminar Proceedings}, ISSN = {1862-4405}, publisher = {Internationales Begegnungs- und Forschungszentrum f{"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2006/445}, annote = {Keywords: Linear programming, rigorous error bounds, netlib, interval arithmetic} }
