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Parametric mixed-integer 0-1 linear programming: The general case for a single parameter

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  • Mitsos, Alexander
  • Barton, Paul I.

Abstract

Two algorithms for the general case of parametric mixed-integer linear programs (MILPs) are proposed. Parametric MILPs are considered in which a single parameter can simultaneously influence the objective function, the right-hand side and the matrix. The first algorithm is based on branch-and-bound on the integer variables, solving a parametric linear program (LP) at each node. The second algorithm is based on the optimality range of a qualitatively invariant solution, decomposing the parametric optimization problem into a series of regular MILPs, parametric LPs and regular mixed-integer nonlinear programs (MINLPs). The number of subproblems required for a particular instance is equal to the number of critical regions. For the parametric LPs an improvement of the well-known rational simplex algorithm is presented, that requires less consecutive operations on rational functions. Also, an alternative based on predictor-corrector continuation is proposed. Numerical results for a test set are discussed.

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  • Mitsos, Alexander & Barton, Paul I., 2009. "Parametric mixed-integer 0-1 linear programming: The general case for a single parameter," European Journal of Operational Research, Elsevier, vol. 194(3), pages 663-686, May.
    • Handle: RePEc:eee:ejores:v:194:y:2009:i:3:p:663-686
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    References listed on IDEAS

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    1. Crema, Alejandro, 1998. "A procedure to verify the completeness of the right-hand-side parametric analysis for a mixed integer linear programming problem," European Journal of Operational Research, Elsevier, vol. 108(3), pages 684-695, August.
    2. Harvey J. Greenberg, 1999. "Matrix Sensitivity Analysis from an Interior Solution of a Linear Program," INFORMS Journal on Computing, INFORMS, vol. 11(3), pages 316-327, August.
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    6. Larry Jenkins, 1982. "Parametric Mixed Integer Programming: An Application to Solid Waste Management," Management Science, INFORMS, vol. 28(11), pages 1270-1284, November.
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    Cited by:

    1. Efstratios Pistikopoulos & Luis Dominguez & Christos Panos & Konstantinos Kouramas & Altannar Chinchuluun, 2012. "Theoretical and algorithmic advances in multi-parametric programming and control," Computational Management Science, Springer, vol. 9(2), pages 183-203, May.
    2. Iosif Pappas & Nikolaos A. Diangelakis & Efstratios N. Pistikopoulos, 2021. "The exact solution of multiparametric quadratically constrained quadratic programming problems," Journal of Global Optimization, Springer, vol. 79(1), pages 59-85, January.
    3. Cristina Bazgan & Arne Herzel & Stefan Ruzika & Clemens Thielen & Daniel Vanderpooten, 2022. "An approximation algorithm for a general class of parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 43(5), pages 1328-1358, July.
    4. Richard Oberdieck & Martina Wittmann-Hohlbein & Efstratios Pistikopoulos, 2014. "A branch and bound method for the solution of multiparametric mixed integer linear programming problems," Journal of Global Optimization, Springer, vol. 59(2), pages 527-543, July.

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