Mathematical Programming

, Volume 99, Issue 2, pp 351–376 | Cite as

Adjustable robust solutions of uncertain linear programs

  • A. Ben-Tal
  • A. Goryashko
  • E. Guslitzer
  • A. Nemirovski
Article

Abstract

We consider linear programs with uncertain parameters, lying in some prescribed uncertainty set, where part of the variables must be determined before the realization of the uncertain parameters (‘‘non-adjustable variables’’), while the other part are variables that can be chosen after the realization (‘‘adjustable variables’’). We extend the Robust Optimization methodology ([1, 3-6, 9, 13, 14]) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterpart (RC), however, in most cases the ARC is computationally intractable (NP-hard). This difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data. The ensuing Affinely Adjustable Robust Counterpart (AARC) problem is then shown to be, in certain important cases, equivalent to a tractable optimization problem (typically an LP or a Semidefinite problem), and in other cases, having a tight approximation which is tractable. The AARC approach is illustrated by applying it to a multi-stage inventory management problem.

Keywords

Uncertain linear programs robust optimization conic optimization semidefinite programming NP-hard continuous optimization problems adjustable robust counterpart affinely-adjustable robust counterpart 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ben-Tal, A., El~Ghaoui, L., Nemirovski, A.: ‘‘Robust Semidefinite Programming.’’ In: R. Saigal, H. Wolkowitcz, L. Vandenberghe, (eds.), Handbook on Semidefinite Programming, Kluwer Academis Publishers, 2000Google Scholar
  2. 2.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2002Google Scholar
  3. 3.
    Ben-Tal, A. Nemirovski, A.: ‘‘Robust Convex Optimization.’’ Math. Oper. Res. 23, (1998)Google Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: ‘‘Robust solutions to uncertain linear programs.’’ OR Letters 25, 1–13 (1999)CrossRefMATHGoogle Scholar
  5. 5.
    Ben-Tal, A., Nemirovski, A.: ‘‘Stable Truss Topology Design via Semidefinite Programming.’’ SIAM J. Optim. 7, 991–1016 (1997)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Ben-Tal, A., Nemirovski, A., Roos, C.: ‘‘Robust solutions of uncertain quadratic and conic-quadratic problems.’’ to appear in SIAM J. on Optimization, 2001Google Scholar
  7. 7.
    Bonhenblust, H.F., Karlin, S., Shapley, L.S.: ‘‘Games with continuous pay-offs.’’ In: Annals of Mathematics Studies, 24, 1950, pp. 181–192Google Scholar
  8. 8.
    Boyd, S., El~Ghaoui, L., Feron, E., Balakrishnan, V.: ‘‘Linear Matrix Inequalities in System and Control Theory.’’ Volume 15 of Studies in Applied Mathematics, SIAM, Philadelphia, 1994Google Scholar
  9. 9.
    Chandrasekaran, S., Golub, G.H., Gu, M., Sayed, A.H.: ‘‘Parameter estimation in the presence of bounded data uncertainty.’’ J. Matrix Anal. Appl. 19, 235–252 (1998)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Dantzig, G.B., Madansky, A.: ‘‘On the Solution of Two-Stage Linear Programs under Uncertainty.’’ Proceedings of the Fourth Berkley Symposium on Statistics and Probability, 1, University California Press, Berkley, CA, 1961, pp. 165–176Google Scholar
  11. 11.
    Grötschel, M., Lovasz, L., Schrijver, A.: ‘‘The Ellipsoid Method and Combinatorial Optimization.’’ Springer, Heidelberg, 1988Google Scholar
  12. 12.
    Guslitser, E.: ‘‘Uncertatinty-immunized solutions in linear programming.’’ Master Thesis, Technion, Israeli Institute of Technology, IE&M faculty 2002. http://iew3.technion.ac.il/Labs/Opt/index.php?4Google Scholar
  13. 13.
    El-Ghaoui, L., Lebret, H.: ‘‘Robust solutions to least-square problems with uncertain data matrices.’’ SIAM J. Matrix Anal. Appl. 18, 1035–1064 (1997)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    El-Ghaoui, L., Oustry, F., Lebret, h.: ‘‘Robust solutions to uncertain semidefinite programs.’’ SIAM J. Optimization 9, 33–52 (1998)CrossRefMATHGoogle Scholar
  15. 15.
    Motskin, T.S.: ‘‘Signs of Minors.’’ Academic Press 1967, pp. 225–240Google Scholar
  16. 16.
    Murty, K.G.: ‘‘Some NP-Complete problems in quadratic and nonlinear programming.’’ Math. Program. 39, 117–129 (1987)MathSciNetMATHGoogle Scholar
  17. 17.
    Prekopa, A.: ‘‘Stochastic Programming.’’ Klumer Academic Publishers, Dordrecht, 1995Google Scholar
  18. 18.
    Soyster, A.L.: ‘‘Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming.’’ Oper. Res. 1154–1157 (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • A. Ben-Tal
    • 1
  • A. Goryashko
    • 1
  • E. Guslitzer
    • 2
  • A. Nemirovski
    • 1
  1. 1.Minerva Optimization Center, Faculty of Industrial Engineering and ManagementTechnionIsrael
  2. 2.Graduate Business SchoolStanford UniversityStanfordUSA

Personalised recommendations