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A parametric simplex algorithm for linear vector optimization problems

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Abstract

In this paper, a parametric simplex algorithm for solving linear vector optimization problems (LVOPs) is presented. This algorithm can be seen as a variant of the multi-objective simplex (the Evans–Steuer) algorithm (Math Program 5(1):54–72, 1973). Different from it, the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions. Instead, it finds a solution in the sense of Löhne (Vector optimization with infimum and supremum. Springer, Berlin, 2011), that is, it finds a subset of efficient solutions that allows to generate the whole efficient frontier. In that sense, it can also be seen as a generalization of the parametric self-dual simplex algorithm, which originally is designed for solving single objective linear optimization problems, and is modified to solve two objective bounded LVOPs with the positive orthant as the ordering cone in Ruszczyński and Vanderbei (Econometrica 71(4):1287–1297, 2003). The algorithm proposed here works for any dimension, any solid pointed polyhedral ordering cone C and for bounded as well as unbounded problems. Numerical results are provided to compare the proposed algorithm with an objective space based LVOP algorithm [Benson’s algorithm in Hamel et al. (J Global Optim 59(4):811–836, 2014)], that also provides a solution in the sense of Löhne (2011), and with the Evans–Steuer algorithm (1973). The results show that for non-degenerate problems the proposed algorithm outperforms Benson’s algorithm and is on par with the Evans–Steuer algorithm. For highly degenerate problems Benson’s algorithm (Hamel et al. 2014) outperforms the simplex-type algorithms; however, the parametric simplex algorithm is for these problems computationally much more efficient than the Evans–Steuer algorithm.

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Acknowledgments

We would like to thank Andreas Löhne, Friedrich-Schiller-Universität Jena, for helpful remarks that greatly improved the manuscript, and Ralph E. Steuer, University of Georgia, for providing us the ADBASE implementation of the algorithm from [15]. Vanderbei’s research was supported by the Office of Naval Research under Award Number N000141310093 and N000141612162.

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Authors and Affiliations

  1. Institute for Statistics and Mathematics, Vienna University of Economics and Business, 1020, Vienna, Austria

    Birgit Rudloff

  2. Department of Industrial Engineering, Bilkent University, 06800, Ankara, Turkey

    Firdevs Ulus

  3. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, 08544, USA

    Robert Vanderbei

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  1. Birgit Rudloff
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  2. Firdevs Ulus
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  3. Robert Vanderbei
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Correspondence to Firdevs Ulus.

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Rudloff, B., Ulus, F. & Vanderbei, R. A parametric simplex algorithm for linear vector optimization problems. Math. Program. 163, 213–242 (2017). https://doi.org/10.1007/s10107-016-1061-z

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