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Multiparametric Linear Programming

Author

Listed:
  • Tomas Gal

    (Technische Hochschule, Aachen, West Germany)

  • Josef Nedoma

    (Czechoslovak Academy of Sciences, Prague, Czechoslovakia)

Abstract

The multiparametric linear programming (MLP) problem for the right-hand sides (RHS) is to maximize z = c T x subject to Ax = b(\lambda), x \geqq 0, where b(\lambda) be expressed in the form where F is a matrix of constant coefficients, and \lambda is a vector-parameter. The multiparametric linear programming (MLP) problem for the prices or objective function coefficients (OFC) is to maximize z = c T (v)x subject to Ax = b, x \geqq 0, where c(I) can be expressed in the form c(v) = c* + Hv, and where H is a matrix of constant coefficients, and v a vector-parameter. Let B i be an optimal basis to the MLP-RHS problem and R i be a region assigned to B i such that for all \lambda \epsilon R i the basis B i is optimal. Let K denote a region such that K = U i R i provided that the R i for various I do not overlap. The purpose of this paper is to present an effective method for finding all regions R i that cover K and do not overlap. This method uses an algorithm that finds all nodes of a finite connected graph. This method uses an algorithm that finds all nodes of a finite connected graph. An analogus method is presented for the MLP-OFC problem.

Suggested Citation

  • Tomas Gal & Josef Nedoma, 1972. "Multiparametric Linear Programming," Management Science, INFORMS, vol. 18(7), pages 406-422, March.
  • Handle: RePEc:inm:ormnsc:v:18:y:1972:i:7:p:406-422
    DOI: 10.1287/mnsc.18.7.406
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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. Goldlücke, Susanne & Kranz, Sebastian, 2012. "Infinitely repeated games with public monitoring and monetary transfers," Journal of Economic Theory, Elsevier, vol. 147(3), pages 1191-1221.
    3. Patrice Gaillardetz & Saeb Hachem, 2019. "Risk-Control Strategies," Papers 1908.02228, arXiv.org.
    4. Benson, Harold P. & Sun, Erjiang, 2002. "A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program," European Journal of Operational Research, Elsevier, vol. 139(1), pages 26-41, May.
    5. Filippi, Carlo & Romanin-Jacur, Giorgio, 2002. "Multiparametric demand transportation problem," European Journal of Operational Research, Elsevier, vol. 139(2), pages 206-219, June.
    6. Cai, Tianxing & Zhao, Chuanyu & Xu, Qiang, 2012. "Energy network dispatch optimization under emergency of local energy shortage," Energy, Elsevier, vol. 42(1), pages 132-145.
    7. 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.
    8. Inuiguchi, Masahiro & Sakawa, Masatoshi, 1995. "Minimax regret solution to linear programming problems with an interval objective function," European Journal of Operational Research, Elsevier, vol. 86(3), pages 526-536, November.
    9. Charitopoulos, Vassilis M. & Dua, Vivek, 2017. "A unified framework for model-based multi-objective linear process and energy optimisation under uncertainty," Applied Energy, Elsevier, vol. 186(P3), pages 539-548.
    10. C. Filippi, 2004. "An Algorithm for Approximate Multiparametric Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 73-95, January.
    11. Amir Akbari & Paul I. Barton, 2018. "An Improved Multi-parametric Programming Algorithm for Flux Balance Analysis of Metabolic Networks," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 502-537, August.
    12. Ilbin Lee & Stewart Curry & Nicoleta Serban, 2019. "Solving Large Batches of Linear Programs," INFORMS Journal on Computing, INFORMS, vol. 31(2), pages 302-317, April.
    13. J. Spjøtvold & P. Tøndel & T. A. Johansen, 2007. "Continuous Selection and Unique Polyhedral Representation of Solutions to Convex Parametric Quadratic Programs," Journal of Optimization Theory and Applications, Springer, vol. 134(2), pages 177-189, August.
    14. Stephan Helfrich & Arne Herzel & Stefan Ruzika & Clemens Thielen, 2022. "An approximation algorithm for a general class of multi-parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1459-1494, October.
    15. 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.
    16. F. Borrelli & A. Bemporad & M. Morari, 2003. "Geometric Algorithm for Multiparametric Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 118(3), pages 515-540, September.
    17. Martina Wittmann-Hohlbein & Efstratios Pistikopoulos, 2013. "On the global solution of multi-parametric mixed integer linear programming problems," Journal of Global Optimization, Springer, vol. 57(1), pages 51-73, September.
    18. Throsby, C.D., 1973. "New Methodologies in Agricultural Production Economics: a Review," 1973 Conference, August 19-30, 1973, São Paulo, Brazil 181385, International Association of Agricultural Economists.
    19. Fabian J. Sting & Arnd Huchzermeier, 2012. "Dual sourcing: Responsive hedging against correlated supply and demand uncertainty," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(1), pages 69-89, February.
    20. Styliani Avraamidou & Efstratios N. Pistikopoulos, 2019. "Multi-parametric global optimization approach for tri-level mixed-integer linear optimization problems," Journal of Global Optimization, Springer, vol. 74(3), pages 443-465, July.

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