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Efficient Storage of Pareto Points in Biobjective Mixed Integer Programming

Author

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  • Nathan Adelgren

    (Department of Mathematics and Computer Science, Edinboro University of Pennsylvania, Edinboro, Pennsylvania 16444)

  • Pietro Belotti

    (FICO, Birmingham B37 7GN, United Kingdom)

  • Akshay Gupte

    (Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634)

Abstract

In biobjective mixed integer linear programs (BOMILPs), two linear objectives are minimized over a polyhedron while restricting some of the variables to be integer. Since many of the techniques for finding or approximating the Pareto set of a BOMILP use and update a subset of nondominated solutions, it is highly desirable to efficiently store this subset. We present a new data structure, a variant of a binary tree that takes as input points and line segments in ℝ 2 and stores the nondominated subset of this input. When used within an exact solution procedure, such as branch and bound (BB), at termination this structure contains the set of Pareto optimal solutions. We compare the efficiency of our structure in storing solutions to that of a dynamic list, which updates via pairwise comparison. Then we use our data structure in two biobjective BB techniques available in the literature and solve three classes of instances of BOMILP, one of which is generated by us. The first experiment shows that our data structure handles up to 10 7 points or segments much more efficiently than a dynamic list. The second experiment shows that our data structure handles points and segments much more efficiently than a list when used in a BB.

Suggested Citation

  • Nathan Adelgren & Pietro Belotti & Akshay Gupte, 2018. "Efficient Storage of Pareto Points in Biobjective Mixed Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 30(2), pages 324-338, May.
  • Handle: RePEc:inm:orijoc:v:30:y:2018:i:2:p:324-338
    DOI: 10.1287/ijoc.2017.0783
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    References listed on IDEAS

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    1. Jesús A. De Loera & Raymond Hemmecke & Matthias Köppe, 2009. "Pareto Optima of Multicriteria Integer Linear Programs," INFORMS Journal on Computing, INFORMS, vol. 21(1), pages 39-48, February.
    2. Gülseren Kiziltan & Erkut Yucaou{g}lu, 1983. "An Algorithm for Multiobjective Zero-One Linear Programming," Management Science, INFORMS, vol. 29(12), pages 1444-1453, December.
    3. Minghe Sun, 2006. "A primogenitary linked quad tree data structure and its application to discrete multiple criteria optimization," Annals of Operations Research, Springer, vol. 147(1), pages 87-107, October.
    4. Ted Ralphs & Matthew Saltzman & Margaret Wiecek, 2006. "An improved algorithm for solving biobjective integer programs," Annals of Operations Research, Springer, vol. 147(1), pages 43-70, October.
    5. Thomas Stidsen & Kim Allan Andersen & Bernd Dammann, 2014. "A Branch and Bound Algorithm for a Class of Biobjective Mixed Integer Programs," Management Science, INFORMS, vol. 60(4), pages 1009-1032, April.
    6. Nicolas Jozefowiez & Gilbert Laporte & Frédéric Semet, 2012. "A Generic Branch-and-Cut Algorithm for Multiobjective Optimization Problems: Application to the Multilabel Traveling Salesman Problem," INFORMS Journal on Computing, INFORMS, vol. 24(4), pages 554-564, November.
    7. H. W. Lenstra, 1983. "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 538-548, November.
    8. Minghe Sun & Ralph E. Steuer, 1996. "Quad-Trees and Linear Lists for Identifying Nondominated Criterion Vectors," INFORMS Journal on Computing, INFORMS, vol. 8(4), pages 367-375, November.
    9. Francis Sourd & Olivier Spanjaard, 2008. "A Multiobjective Branch-and-Bound Framework: Application to the Biobjective Spanning Tree Problem," INFORMS Journal on Computing, INFORMS, vol. 20(3), pages 472-484, August.
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    Cited by:

    1. Sophie N. Parragh & Fabien Tricoire, 2019. "Branch-and-Bound for Bi-objective Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 31(4), pages 805-822, October.

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