On the Relationship between $\varepsilon$-sensitivity Analysis and Sensitivity Analysis using an Optimal Basis

  • Park, Chan-Kyoo (Dept. of Management, Dongguk University) ;
  • Kim, Woo-Je (Dept. of Industrial and Information Systems Engienering, Seoul National University) ;
  • Park, Soondal (Dept. of Industrial Engineering, Seoul National Univ.)
  • Published : 2004.11.01


$\epsilon$-sensitivity analysis is a kind of methods for performing sensitivity analysis for linear programming. Its main advantage is that it can be directly applied for interior-point methods with a little computation. Although $\epsilon$-sensitivity analysis was proposed several years ago, there have been no studies on its relationship with other sensitivity analysis methods. In this paper, we discuss the relationship between $\epsilon$-sensitivity analysis and sensitivity analysis using an optimal basis. First. we present a property of $\epsilon$-sensitivity analysis, from which we derive a simplified formula for finding the characteristic region of $\epsilon$-sensitivity analysis. Next, using the simplified formula, we examine the relationship between $\epsilon$-sensitivity analysis and sensitivity analysis using optimal basis when an $\epsilon$-optimal solution is sufficiently close to an optimal extreme solution. We show that under primal nondegeneracy or dual non degeneracy of an optimal extreme solution, the characteristic region of $\epsilon$-sensitivity analysis converges to that of sensitivity analysis using an optimal basis. However, for the case of both primal and dual degeneracy, we present an example in which the characteristic region of $\epsilon$-sensitivity analysis is different from that of sensitivity analysis using an optimal basis.



  1. Adler, I. and R. D. C. Monteiro, 'A geometric view of parametric linear programming,' Algorithmica 8 (1992), 161-176 https://doi.org/10.1007/BF01758841
  2. Bixby, R. E. and M. J. Saltzman, 'Recovering an optimal LP basis from an interior point solution,' Operations Research Letters 15 (1994), 169-178 https://doi.org/10.1016/0167-6377(94)90074-4
  3. Dantzig, G. B., Linear Programming and Extension, Princeston University Press, 1963
  4. Park, C.-K, W.-J. Kim, S. Park, 'Generalized sensitivity analysis at a degenerate optimal solution,' Journal of the Korean Operations Research and Management Science Society 25, 4 (2000), 1-14
  5. Park, C.-K, W.-J. Kim, S. Lee, S. Park, 'Positive sensitivity analysis in linear programming,' Asia-Pacific Journal of Operational Research, 2004, To appear https://doi.org/10.1142/S0217595904000059
  6. Gal, T., Postoptimal analyses, parametric programming, and related topics, MacGraw-Hill, New York, 1979
  7. Golub, G. H. and C. F. Van Loan, Matrix Computations, John Hopkins University Press, 1983
  8. Jansen, B., J. J. de Jong, C. Roos, T. Terlaky, 'Sensitivity analysis in linear programming: just be careful!,' European Journal of Operational Research 101 (1997), 15-28 https://doi.org/10.1016/S0377-2217(96)00172-5
  9. Kim, W. J., C.-K. Park, S. Park, 'An e-sensitivity analysis in the primaldual interior point method,' European Journal of Operational Research 116 (1999), 629-639 https://doi.org/10.1016/S0377-2217(98)00117-9
  10. Mehrotra, S. and Y. Ye, 'Finding an interior point in the optimal face of linear programs,' Mathematical Programming 62 (1993), 497-515 https://doi.org/10.1007/BF01585180
  11. Stewart, G. W., 'On scaled projections and pseudo inverses,' Linear Algebra and its Applications 112 (1989), 189-193 https://doi.org/10.1016/0024-3795(89)90594-6
  12. Tapia, R. A. and Y. Zhang, 'An optimal-basis identification technique for interior-point linear programming algorithms,' Linear Algebra and Its Applications 152 (1991), 343-363 https://doi.org/10.1016/0024-3795(91)90281-Z
  13. Yang, B. H., 'A study on sensitivity analysis for a non-extreme optimal solution in linear programming,' Ph. D. Dissertation, Seoul National University, Republic of Korea, 1990