TY - JOUR

T1 - A parametric simplex algorithm for linear vector optimization problems

AU - Rudloff, Birgit

AU - Ulus, Firdevs

AU - Vanderbei, Robert

N1 - Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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.

AB - 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.

KW - Algorithms

KW - Linear vector optimization

KW - Multiple objective optimization

KW - Parameter space segmentation

UR - http://www.scopus.com/inward/record.url?scp=84981161296&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84981161296&partnerID=8YFLogxK

U2 - 10.1007/s10107-016-1061-z

DO - 10.1007/s10107-016-1061-z

M3 - Article

AN - SCOPUS:84981161296

SN - 0025-5610

VL - 163

SP - 213

EP - 242

JO - Mathematical Programming

JF - Mathematical Programming

IS - 1-2

ER -