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Linear Superiorization for Infeasible Linear Programming

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Discrete Optimization and Operations Research (DOOR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9869))

Abstract

Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibility-seeking iterative process toward feasible points that have lower (not necessarily minimal) values of the objective function than points that would have been reached by the same feasiblity-seeking iterative process without superiorization. Using a feasibility-seeking iterative process that converges even if the linear feasible set is empty, LinSup generates an iterative sequence that converges to a point that minimizes a proximity function which measures the linear constraints violation. In addition, due to LinSup’s repeated objective function reduction steps such a point will most probably have a reduced objective function value. We present an exploratory experimental result that illustrates the behavior of LinSup on an infeasible LP problem.

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Notes

  1. 1.

    Some support for this reasoning may be borrowed from the American scientist and Noble-laureate Herbert Simon who was in favor of “satisficing” rather then “maximizing”. Satisficing is a decision-making strategy that aims for a satisfactory or adequate result, rather than the optimal solution. This is because aiming for the optimal solution may necessitate needless expenditure of time, energy and resources. The term “satisfice” was coined by Herbert Simon in 1956 [20], see: https://en.wikipedia.org/wiki/Satisficing.

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Acknowledgments

We thank Gabor Herman, Ming Jiang and Evgeni Nurminski for reading a previous version of the paper and sending us comments that helped improve it. This work was supported by Research Grant No. 2013003 of the United States-Israel Binational Science Foundation (BSF).

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Authors and Affiliations

  1. Department of Mathematics, University of Haifa, Mt. Carmel, 3498838, Haifa, Israel

    Yair Censor & Yehuda Zur

Authors
  1. Yair Censor
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  2. Yehuda Zur
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Corresponding author

Correspondence to Yair Censor .

Editor information

Editors and Affiliations

  1. Sobolev Institute of Mathematics , Novosibirsk, Russia

    Yury Kochetov

  2. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  3. Sobolev Institute of Mathematics , Novosibirsk, Russia

    Vladimir Beresnev

  4. Far Eastern Federal University , Vladivostik, Russia

    Evgeni Nurminski

  5. University of Florida , Gainesville, Florida, USA

    Panos Pardalos

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Censor, Y., Zur, Y. (2016). Linear Superiorization for Infeasible Linear Programming. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds) Discrete Optimization and Operations Research. DOOR 2016. Lecture Notes in Computer Science(), vol 9869. Springer, Cham. https://doi.org/10.1007/978-3-319-44914-2_2

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  • DOI: https://doi.org/10.1007/978-3-319-44914-2_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-44913-5

  • Online ISBN: 978-3-319-44914-2

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