Bi-parametric optimal partition invariancy sensitivity analysis in linear optimization

  • Alireza Ghaffari-Hadigheh
  • Habib Ghaffari-Hadigheh
  • Tamás Terlaky
Original Paper

Abstract

In bi-parametric linear optimization (LO), perturbation occurs in both the right-hand-side and the objective function data with different parameters. In this paper, the bi-parametric LO problem is considered and we are interested in identifying the regions where the optimal partitions are invariant. These regions are referred to as invariancy regions. It is proved that invariancy regions are separated by vertical and horizontal lines and generate a mesh-like area. It is proved that the boundaries of these regions can be identified in polynomial time. The behavior of the optimal value function on these regions is investigated too.

Keywords

Linear optimization Bi-parametric sensitivity analysis Optimal partition Invariancy region Optimal value function 

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References

  1. Adler I and Monteiro R (1992). A geomertic view of parametric linear programing. Algorithmica 8: 161–176 CrossRefGoogle Scholar
  2. Dantzig GB (1963). Linear programming and extensions. Princeton University Press, Princeton Google Scholar
  3. Gal T and Greenberg HJ (1997). Advances in sensitivity analysis and parametric programming. Kluwer, London Google Scholar
  4. Ghaffari-Hadigheh A, Romanko O and Terlaky T (2007). Sensitivity analysis in convex quadratic optimization: simultaneous perturbation of the objective and right-hand-side vectors. Algoritmic Oper Res 2: 94–111 Google Scholar
  5. Goldman AJ and Tucker AW (1956). Theory of linear programming. In: Kuhn, HW and Tucker, AW (eds) Linear inequalities and related systems annals of mathematical studies 38. pp 63–97. Princeton University Press, Princeton Google Scholar
  6. Guddat J, Vasquez FG, Tammer K and Wendler K (1985). Multiobjective and stochastic optimization based on parametric optimization. Akademie, Berlin Google Scholar
  7. Güler O and Ye Y (1993). Convergence behavior of interior-point algorithms. Math Program 60(2): 215–228 CrossRefGoogle Scholar
  8. Hollatz H and Weinert H (1971). Ein Algorithums zur Lösung des doppelt-einparametrischen linearen Optimierungsproblems. Math Oper Stat 2: 181–197 Google Scholar
  9. Illés T, Peng J, Roos C and Terlaky T (2000). A strongly polynomial rounding procedure yielding a maximally complementary solution for P *(κ) linear complementarity problems. SIAM J Optim 11(2): 320–340 CrossRefGoogle Scholar
  10. Jansen B, Roos C, Terlaky T (1984) An interior point approach to postoptimal and parametric analysis in linear programming, Report No. 92-90, Faculty of Technical Mathematics and Computer Science, Delft University of Technology, Delft, The NetherlandsGoogle Scholar
  11. Karmarkar NK (1984). A new polynomial-time algorithm for linear programming. Combinatorica 4: 375–395 CrossRefGoogle Scholar
  12. Mehrotra S and Ye Y (1993). Finding an interior-point in the optimal face of linear programs. Math Program 62(3): 497–515 CrossRefGoogle Scholar
  13. Nožička F (1972). Über eine Klasse von linearen einparametrischen Optimierungsproblemen. Math Oper Stat 3: 159–194 Google Scholar
  14. Nožička F, Guddat J, Hollatz H and Bank B (1974). Theorie der linearen parametrischen optimierung. Akademie, Berlin Google Scholar
  15. Roos C, Terlaky T and Vial J-Ph (2005). Interior point algorithms for linear optimization. Springer, Boston Google Scholar
  16. Weinert H (1970). Doppelt-einparametrische lineare Optimierung. I: Unabhängige Parameter. Math Optim Stat 1: 173–197 Google Scholar
  17. Ye Y (1992). On the finite convergence of interior-point algorithms for linear programming. Math Program 57(2): 325–335 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Alireza Ghaffari-Hadigheh
    • 1
  • Habib Ghaffari-Hadigheh
    • 2
  • Tamás Terlaky
    • 3
  1. 1.Department of MathematicsAzarbaijan University of Tarbiat MoallemTabrizIran
  2. 2.Department of MathematicsPayame noor UniversityShabestarIran
  3. 3.School of Computational Engineering and Science, Department of Computing and SoftwareMcMaster UniversityHamiltonCanada

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