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Linear programming using limited-precision oracles

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Abstract

Since the elimination algorithm of Fourier and Motzkin, many different methods have been developed for solving linear programs. When analyzing the time complexity of LP algorithms, it is typically either assumed that calculations are performed exactly and bounds are derived on the number of elementary arithmetic operations necessary, or the cost of all arithmetic operations is considered through a bit-complexity analysis. Yet in practice, implementations typically use limited-precision arithmetic. In this paper we introduce the idea of a limited-precision LP oracle and study how such an oracle could be used within a larger framework to compute exact precision solutions to LPs. Under mild assumptions, it is shown that a polynomial number of calls to such an oracle and a polynomial number of bit operations, is sufficient to compute an exact solution to an LP. This work provides a foundation for understanding and analyzing the behavior of the methods that are currently most effective in practice for solving LPs exactly.

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Notes

  1. Otherwise, as noted in [16], an oracle that guarantees a solution with residual errors bounded by a fixed constant could be queried for solutions of arbitrarily high precision in a single call by first scaling the problem data by a large constant. As we will see, this type of manipulation is not used by our algorithms. Moreover, the requirement in the definition of the oracle that the input data entries are in the bounded set \(\mathbb {F}(p)\) would forbid the input of arbitrarily scaled LPs.

  2. In order to account fairly for small arbitrary deviations in time measurements, an algorithm is considered “fastest” if it solves an instance in no more than 5% of the time taken by the fastest method. Hence, the numbers do not add up to the total of 1202 instances. Furthermore, note that the same picture holds also when excluding easy instances that took less than 2 s by all solvers: SoPlex\(_{\text {fac}}\) wins more than twice as often as QSopt_ex.

  3. On three instances, floating-point SoPlex could not solve the first refinement LP within the time limit. For three further instances, the initial floating-point solve incorrectly claimed unboundedness and for one instance it incorrectly claimed infeasibility; in all of these cases, the incorrect claims were rejected successfully using feasibility and unboundedness tests as described in [16], but after starting to refine the original LP again, floating-point SoPlex  failed to return an approximately optimal solution even when trying to run with different floating-point settings.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their detailed study of the manuscript and their comments, which were of exceptionally high quality.

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  1. Konrad-Zuse-Zentrum für Informationstechnik Berlin, Takustr. 7, 14195, Berlin, Germany

    Ambros Gleixner

  2. Mathematics and Statistics, Oakland University, Rochester, Michigan, 48309, USA

    Daniel E. Steffy

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  1. Ambros Gleixner
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Correspondence to Daniel E. Steffy.

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An extended abstract of this work appeared as: Gleixner A., Steffy D.E. (2019) Linear Programming Using Limited-Precision Oracles. In: Lodi A., Nagarajan V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2019. Lecture Notes in Computer Science, vol 11480. Springer, Cham.

Ambros Gleixner was supported by the Research Campus MODAL Mathematical Optimization and Data Analysis Laboratories funded by the German Ministry of Education and Research (BMBF Grant Number 05M14ZAM).

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Gleixner, A., Steffy, D.E. Linear programming using limited-precision oracles. Math. Program. 183, 525–554 (2020). https://doi.org/10.1007/s10107-019-01444-6

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