Journal of Global Optimization

, Volume 70, Issue 4, pp 811–842 | Cite as

A Utility Theory Based Interactive Approach to Robustness in Linear Optimization

Article
  • 85 Downloads

Abstract

We treat uncertain linear programming problems by utilizing the notion of weighted analytic centers and notions from the area of multi-criteria decision making. After introducing our approach, we develop interactive cutting-plane algorithms for robust optimization, based on concave and quasi-concave utility functions. In addition to practical advantages, due to the flexibility of our approach, we prove that under a theoretical framework proposed by Bertsimas and Sim (Oper Res 52:35–53, 2004), which establishes the existence of certain convex formulation of robust optimization problems, the robust optimal solutions generated by our algorithms are at least as desirable to the decision maker as any solution generated by many other robust optimization algorithms in the theoretical framework. We present some probabilistic bounds for feasibility of robust solutions and evaluate our approach by means of computational experiments.

Keywords

Linear optimization Uncertainty in data Robust optimization Cutting-plane algorithms Weighted analytic centers Interactive decision making 

References

  1. 1.
    Anderson, J., Jibrin, S.: An interior point method for linear programming using weighted analytic centers. J. Ariz. Nev. Acad. Sci. 41(1), 1–7 (2009)CrossRefGoogle Scholar
  2. 2.
    Anstreicher, K.M.: On Vaidya’s volumetric cutting plane method for convex programming. Math. Oper. Res. 22, 63–89 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ardel, A., Oren, S.: Using approximate gradients in developing an interactive interior primal-dual multiobjective linear programming algorithm. Eur. J. Oper. Res. 89, 202–211 (1996)CrossRefMATHGoogle Scholar
  4. 4.
    Atkinson, D.S., Vaidya, P.M.: A cutting plane algorithm for convex programming that uses analytic centers. Math. Program. 69(1-3), 1–43 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ben-Tal, A., Boyd, S., Nemirovski, A.: Extending scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math. Program. 107, 63–89 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robustness Optimization. Princeton University Press, Princeton (2009)MATHGoogle Scholar
  7. 7.
    Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Program. 99, 351–376 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88, 411–424 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bertsimas, D., Nohadani, O.: Robust optimization with simulated annealing. J. Global Optim. 48(2), 323–334 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bertsimas, D., Pachamanova, D., Sim, M.: Robust linear optimization under general norms. Oper. Res. Lett. 32, 510–516 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory—a convex optimization approach. SIAM J. Optim. 15, 780–804 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bertsimas, D., Sim, M.: Tractable approximations to robust conic optimization problems. Math. Program. 107, 5–36 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52, 35–53 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Math. Program. 98, 49–71 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefMATHGoogle Scholar
  18. 18.
    Bortfeld, T., Chan, T.C.Y., Trofimov, A., Tsitsiklis, J.N.: Robust management of motion uncertainty in intensity-modulated radiation therapy. Oper. Res. 56, 1461–1473 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Boyd, S., Vanderberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  20. 20.
    Chan, T.C., Mišić, V.V.: Adaptive and robust radiation therapy optimization for lung cancer. Eur. J. Oper. Res. 231, 745–756 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Chu, M., Zinchenko, Y., Henderson, S.G., Sharpe, M.B.: Robust optimization for intensity modulated radiation therapy treatment planning under uncertainty. Phys. Med. Biol. 50, 5463–5477 (2006)CrossRefGoogle Scholar
  22. 22.
    Coco, A.A., Júnior, J.C.A., Noronha, T.F., Santos, A.C.: An integer linear programming formulation and heuristics for the minmax relative regret robust shortest path problem. J. Global Optim. 60(2), 265–287 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    El Ghaoui, L., Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9, 33–52 (1998)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. 107, 37–90 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Goffin, J.L., Luo, Z.Q., Ye, Y. : On the complexity of a column generation algorithm for convex and quasiconvex feasibility problems. In: Large Scale Optimization: State of the Art, pp. 187–196. Kluwer, Dordrecht (1993)Google Scholar
  26. 26.
    Goffin, J.L., Vial, J.P.: Convex non-differentiable optimization: a survey focused on the analytic center cutting-plane method. Optim. Methods Softw. 17, 805–867 (2002)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58(4-part-1), 902–917 (2010)Google Scholar
  28. 28.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Grünbaum, B.: Partitions of mass-distributions and convex bodies by hyperplanes. Pac. J. Math. 10, 1257–1261 (1960)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hanasusanto, G.A., Roitch, V., Kuhn, D., Wiesemann, W.: A distributionally robust perspective on uncertainty quantification and chance constrained programming. Math. Program. 151(1), 35–62 (2015)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Hu, J., Mehrotra, S.: Robust and stochastically weighted multiobjective optimization models and reformulations. Oper. Res. 60, 936–953 (2012)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Iancu, D.A., Trichakis, N.: Pareto efficiency in robust optimization. Manag. Sci. 60(1), 130–147 (2013)CrossRefGoogle Scholar
  33. 33.
    Ignizio, J.P.: Goal Programming and Extensions. Lexington Books, Lexington (1976)Google Scholar
  34. 34.
    Iyengar, V.S., Lee, J., Campbell, M.: Q-Eval: Evaluating multiple attribute items using queries. In: Proceedings of the 3rd ACM Conference on Electronic Commerce, pp. 144–153 (2001)Google Scholar
  35. 35.
    Kaelbling, L.P., Littman, M.L., Moore, A.W.: Reinforcement learning: a survey. J. Artif. Intell. Res. 4, 237–285 (1996)Google Scholar
  36. 36.
    Karimi, M.: A quick-and-dirty approach to robustness in linear optimization, Master’s Thesis, University of Waterloo, (2012)Google Scholar
  37. 37.
    Karimi, M., Moazeni, S., Tunçel, L.: A utility theory based interactive approach to robustness in linear optimization. arXiv:1312.4489
  38. 38.
    Keeney, R.: Value-Focused Thinking. Harvard University Press, London (1992)MATHGoogle Scholar
  39. 39.
    Keeney, R., Raiffa, H.: Decision with Multiple Objectives. Wiley, New York (1976)MATHGoogle Scholar
  40. 40.
    Khachiyan, L.G.: Polynomial algorithms in linear programming. USSR Comput. Math. Math. Phys. 20(1), 53–72 (1980)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, vol. 538. Springer, New York (1991)MATHGoogle Scholar
  42. 42.
    Köksalan, M., Wallenius, J., Zionts, S.: Multiple Criteria Decision Making: From Early History to the 21st Century. World Scientific, Singapore (2011)CrossRefGoogle Scholar
  43. 43.
    Lu, D., Gzara, F.: The robust crew pairing problem: model and solution methodology. J. Global Optim. 62(1), 29–54 (2015)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Miller, L.B., Wagner, H.: Chance-constrained programming with joint constraints. Oper. Res. 13, 930–945 (1965)CrossRefMATHGoogle Scholar
  45. 45.
    Minoux, M.: On 2-stage robust LP with RHS uncertainty: complexity results and applications. J. Global Optim. 49(3), 521–537 (2011)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Moazeni, S.: Flexible robustness in linear optimization. Master’s Thesis, University of Waterloo (2006)Google Scholar
  47. 47.
    Monteiro, R.D.C., Zanjácomo, P.R.: General interior-point maps and existence of weighted paths for nonlinear semidefinite complementarity problems. Math. Oper. Res. 25(3), 381–399 (2000)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Morgan, M.G., Henrion, M.: Uncertainty—A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis. Cambridge University Press, New York (1990)CrossRefGoogle Scholar
  49. 49.
    Mudchanatongsuk, S., Ordonez, F., Liu, J.: Robust Solutions for Network Design Under Transportation Cost And Demand Uncertainty. USC ISE working paper (2005-05)Google Scholar
  50. 50.
    Mulvey, J.M., Vanderbei, R.J., Zenios, S.A.: Robust optimization of large-scale systems. Oper. Res. 43, 264–281 (1995)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 4, 969–996 (2006)MathSciNetMATHGoogle Scholar
  52. 52.
    Nesterov, Yu.: Complexity estimates of some cutting-plane methods based on the analytic barrier. Math. Program. Ser. B 69, 149–176 (1995)MathSciNetMATHGoogle Scholar
  53. 53.
    Nesterov, Yu., Nemirovskii, A.: Interior Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)CrossRefMATHGoogle Scholar
  54. 54.
    Newman, D.J.: Location of the maximum on unimodal surfaces. JACM 12, 395–398 (1965)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Ordonez, F., Zhao, J.: Robust Capacity Expansion Of Network Flows. USC-ISE working paper (2004-01)Google Scholar
  56. 56.
    Parpas, P., Rustem, B., Pistikopoulos, E.N.: Global optimization of robust chance constrained problems. J. Global Optim. 43(2-3), 231–247 (2009)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1997)MATHGoogle Scholar
  58. 58.
    Santoso, T., Ahmed, S., Goetschalckx, M., Shapiro, A.: A stochastic programming approach for supply chain network design under uncertainty. Eur. J. Oper. Res. 167, 96–115 (2005)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Shapiro, A.: Distributionally robust stochastic programming. Optim. Online (2016). http://www.optimization-online.org/DB_HTML/2015/12/5238.html
  60. 60.
    Sherali, H.D., Ganesan, V.: An inverse reliability-based approach for designing under uncertainty with application to robust piston design. J. Global Optim. 37(1), 47–62 (2007)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Sir, M.Y., Epelman, M.A., Pollock, S.M.: Stochastic programming for off-line adaptive radiotherapy. Ann. Oper. Res. 196, 767–797 (2012)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154–1157 (1973)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Sutton, R.S., Barto, A.G.: Reinforcement learning: an introduction. MIT Press, Cambridge (1998)Google Scholar
  64. 64.
    Vaidya, P.M.: A new algorithm for minimizing convex functions over convex sets. In: Symposium on Foundations of Computer Science, pp. 338–343 (1989)Google Scholar
  65. 65.
    Vaidya, P.M., Atkinson, D.S.: A Technique for Bounding the Number of Iterations in Path Following Algorithms, Complexity in Numerical Optimization, pp. 462–489. World Scientific, Singapore (1993)MATHGoogle Scholar
  66. 66.
    Wang, F., Xu, D., Wu, C.: Combinatorial approximation algorithms for the robust facility location problem with penalties. J. Global Optim. 64, 483–496 (2016)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Yudin, D.B., Nemirovski, A.S.: Informational complexity and efficient methods for solving complex extremal problems. Matekon 13, 25–45 (1977)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.School of Business, Stevens Institute of TechnologyHobokenUSA
  3. 3.Department of Combinatorics and Optimization, Faculty of MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations