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A Brief Overview of Interdiction and Robust Optimization

  • Leonardo LozanoEmail author
  • J. Cole Smith
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 152)

Abstract

Two-player optimization problems span an impressive array of possible situations, including cases in which both players optimize their own objective with no regard for the other’s goals, or in which one agent seeks to impede the other’s objective. The agents may commit their decisions simultaneously, using either deterministic or random (mixed) strategies. Alternatively, they can play them in sequence, where one agent has complete or partial knowledge of the other’s decisions. This overview provides the reader insights and entry points into learning about two-stage zero-sum games (e.g., minimax or maximin) in which one agent has complete knowledge of the other’s actions. The difference between interdiction and robust optimization models is described, with a focus on steering the reader to relevant and contemporary research in the field.

References

  1. 1.
    Atamtürk, A., Zhang, M.: Two-stage robust network flow and design under demand uncertainty. Oper. Res. 55(4), 662–673 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bayrak, H., Bailey, M.D.: Shortest path network interdiction with asymmetric information. Networks 52(3), 133–140 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Tal, A., Boyd, S., Nemirovski, A.: Extending scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math. Program. 107(1–2):63–89 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ben-Tal, A., Do Chung, B., Mandala, S.R., Yao, T.: Robust optimization for emergency logistics planning: risk mitigation in humanitarian relief supply chains. Transp. Res. B: Methodolog. 45(8), 1177–1189 (2011)CrossRefGoogle Scholar
  5. 5.
    Ben-Tal, A., El-Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)zbMATHCrossRefGoogle Scholar
  6. 6.
    Ben-Tal, A., Golany, B., Nemirovski, A., Vial, J.-P.: Retailer-supplier flexible commitments contracts: a robust optimization approach. Manuf. Serv. Oper. Manag. 7(3), 248–271 (2005)CrossRefGoogle Scholar
  7. 7.
    Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Program. 99(2), 351–376 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ben-Tal, A., Nemirovski, A.: Stable truss topology design via semidefinite programming. SIAM J. Optim. 7, 991–1016 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ben-Tal, A., Nemirovski, A.: Robust solutions to uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Ben-Tal, A., Nemirovski, A.: Robust optimization–methodology and applications. Math. Program. 92(3), 453–480 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bertsimas, D., Brown, D.B.: Constructing uncertainty sets for robust linear optimization. Oper. Res. 57(6), 1483–1495 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bertsimas, D., Dunning, I., Lubin, M.: Reformulation versus cutting-planes for robust optimization. Comput. Manag. Sci. 13(2), 195–217 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bertsimas, D., Litvinov, E., Sun, X.A., Zhao, J., Zheng, T.: Adaptive robust optimization for the security constrained unit commitment problem. IEEE Trans. Power Syst. 28(1), 52–63 (2013)CrossRefGoogle Scholar
  15. 15.
    Bertsimas, D., Pachamanova, D., Sim, M.: Robust linear optimization under general norms. Oper. Res. Lett. 32(6), 510–516 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Math. Program. 98, 49–71 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Bertsimas, D., Sim, M.: Tractable approximations to robust conic optimization problems. Math. Program. 107(1–2):5–36 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Bertsimas, D., Sim, M., Zhang, M.: Adaptive distributionally robust optimization. Manag. Sci. 65(2), 604–618 (2019)CrossRefGoogle Scholar
  20. 20.
    Bertsimas, D., Thiele, A.: A robust optimization approach to inventory theory. Oper. Res. 54(1), 150–168 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Brown, G., Carlyle, M., Diehl, D., Kline, J., Wood, K.: A two-sided optimization for theater ballistic missile defense. Oper. Res. 53(5), 745–763 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Brown, G.G., Carlyle, W.M., Harney, R., Skroch, E., Wood, R.K.: Interdicting a nuclear-weapons project. Oper. Res. 57(4), 866–877 (2009)zbMATHCrossRefGoogle Scholar
  23. 23.
    Brown, G.G., Carlyle, W.M., Salmerón, J., Wood, R.K.: Analyzing the vulnerability of critical infrastructure to attack and planning defenses. In: Greenberg, H.J., Smith, J.C. (eds.) Tutorials in Operations Research: Emerging Theory, Methods, and Applications, pp. 102–123. INFORMS, Hanover (2005)Google Scholar
  24. 24.
    Brown, G.G., Carlyle, W.M., Salmerón, J., Wood, R.K.: Defending critical infrastructure. Interfaces 36(6), 530–544 (2006)CrossRefGoogle Scholar
  25. 25.
    Cappanera, P., Scaparra, M.P.: Optimal allocation of protective resources in shortest-path networks. Transp. Sci. 45(1), 64–80 (2011)CrossRefGoogle Scholar
  26. 26.
    Church, R.L., Scaparra, M.P.: The r-interdiction median problem with fortification. Geogr. Anal. 39(2), 129–146 (2007)CrossRefGoogle Scholar
  27. 27.
    Church, R.L., Scaparra, M.P., Middleton, R.S.: Identifying critical infrastructure: the median and covering facility interdiction problems. Ann. Assoc. Am. Geogr. 94(3), 491–502 (2004)CrossRefGoogle Scholar
  28. 28.
    Cormican, K.J., Morton, D.P., Wood, R.K.: Stochastic network interdiction. Oper. Res. 46(2), 184–197 (1998)zbMATHCrossRefGoogle Scholar
  29. 29.
    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Esfahani, P.M., Kuhn, D.: Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Math. Program. 171(1–2), 115–166 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Fischetti, M., Monaci, M., Sinnl, M.: A dynamic reformulation heuristic for generalized interdiction problems. Eur. J. Oper. Res. 267(1), 40–51 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Fulkerson, D.R., Harding, G.C.: Maximizing minimum source-sink path subject to a budget constraint. Math. Program. 13(1), 116–118 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58(4-part-1), 902–917 (2010)Google Scholar
  34. 34.
    Golden, B.: A problem in network interdiction. Nav. Res. Logist. Q. 25(4), 711–713 (1978)zbMATHCrossRefGoogle Scholar
  35. 35.
    Gorissen, B.L., Yanıkoğlu, İ., den Hertog, D.: A practical guide to robust optimization. Omega 53, 124–137 (2015)CrossRefGoogle Scholar
  36. 36.
    Gregory, C., Darby-Dowman, K., Mitra, G.: Robust optimization and portfolio selection: the cost of robustness. Eur. J. Oper. Res. 212(2), 417–428 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Held, H., Hemmecke, R., Woodruff, D.L.: A decomposition algorithm applied to planning the interdiction of stochastic networks. Naval Res. Logist. 52(4), 321–328 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Held, H., Woodruff, D.L.: Heuristics for multi-stage interdiction of stochastic networks. J. Heuristics 11(6), 483–500 (2005)zbMATHCrossRefGoogle Scholar
  39. 39.
    Israeli, E., Wood, R.K.: Shortest-path network interdiction. Networks 40(2), 97–111 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Kouvelis, P., Yu, G.: Robust Discrete Optimization and its Applications, vol. 14. Springer Science & Business Media, Dordrecht (1997)zbMATHCrossRefGoogle Scholar
  41. 41.
    Li, Z., Ding, R., Floudas, C.A.: A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization. Ind. Eng. Chem. Res. 50(18), 10567–10603 (2011)Google Scholar
  42. 42.
    Lim, C., Smith, J.C.: Algorithms for discrete and continuous multicommodity flow network interdiction problems. IIE Trans. 39(1), 15–26 (2007)CrossRefGoogle Scholar
  43. 43.
    Lin, X., Janak, S.L., Floudas, C.A.: A new robust optimization approach for scheduling under uncertainty: I. Bounded uncertainty. Comput. Chem. Eng. 28(6–7), 1069–1085 (2004)CrossRefGoogle Scholar
  44. 44.
    Lorca, Á., Sun, X.A., Litvinov, E., Zheng, T.: Multistage adaptive robust optimization for the unit commitment problem. Oper. Res. 64(1), 32–51 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Lozano, L., Smith, J.C.: A backward sampling framework for interdiction problems with fortification. INFORMS J. Comput. 29(1), 123–139 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Lozano, L., Smith, J.C., Kurz, M.E.: Solving the traveling salesman problem with interdiction and fortification. Oper. Res. Lett. 45(3), 210–216 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Moon, Y., Yao, T.: A robust mean absolute deviation model for portfolio optimization. Comput. Oper. Res. 38(9), 1251–1258 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Morton, D.P., Pan, F., Saeger, K.J.: Models for nuclear smuggling interdiction. IIE Trans. 39(1), 3–14 (2007)CrossRefGoogle Scholar
  49. 49.
    Natarajan, K., Pachamanova, D., Sim, M.: Constructing risk measures from uncertainty sets. Oper. Res. 57(5), 1129–1141 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Pan, F., Charlton, W., Morton, D.P.: Interdicting smuggled nuclear material. In: Woodruff, D.L. (ed.) Network Interdiction and Stochastic Integer Programming, pp. 1–20. Kluwer Academic, Boston (2003)Google Scholar
  51. 51.
    Prince, M., Smith, J.C., Geunes, J.: A three-stage procurement optimization problem under uncertainty. Naval Res. Logist. 60(1), 395–412 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Royset, J.O., Wood, R.K.: Solving the bi-objective maximum-flow network-interdiction problem. INFORMS J. Comput. 19(2), 175–184 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Salmerón, J., Wood, K., Baldick, R.: Analysis of electric grid security under terrorist threat. IEEE Trans. Power Syst. 19(2), 905–912 (2004)CrossRefGoogle Scholar
  54. 54.
    Salmerón, J., Wood, K., Baldick, R.: Worst-case interdiction analysis of large-scale electric power grids. IEEE Trans. Power Syst. 24(1), 96–104 (2009)CrossRefGoogle Scholar
  55. 55.
    Scaparra, M.P., Church, R.L.: A bilevel mixed-integer program for critical infrastructure protection planning. Comput. Oper. Res. 35(6), 1905–1923 (2008)zbMATHCrossRefGoogle Scholar
  56. 56.
    Scaparra, M.P., Church, R.L.: An exact solution approach for the interdiction median problem with fortification. Eur. J. Oper. Res. 189(1), 76–92 (2008)zbMATHCrossRefGoogle Scholar
  57. 57.
    Smith, J.C.: Basic interdiction models. In: Cochran, J. (ed.) Wiley Encyclopedia of Operations Research and Management Science, pp. 323–330. Wiley, Hoboken (2010)Google Scholar
  58. 58.
    Smith, J.C., Lim, C.: Algorithms for network interdiction and fortification games. In: Migdalas, A., Pardalos, P.M., Pitsoulis, L., Chinchuluun, A. (eds.) Pareto Optimality, Game Theory and Equilibria. Nonconvex Optimization and its Applications Series, pp. 609–644. Springer, New York (2008)Google Scholar
  59. 59.
    Smith, J.C., Lim, C., Sudargho, F.: Survivable network design under optimal and heuristic interdiction scenarios. J. Glob. Optim. 38(2), 181–199 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Thiele, A., Terry, T., Epelman, M.: Robust linear optimization with recourse. Technical Report, Lehigh University, Bethlehem, PA (2009)Google Scholar
  61. 61.
    Washburn, A., Wood, R.K.: Two-person zero-sum games for network interdiction. Oper. Res. 43(2), 243–251 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Wollmer, R.D.: Removing arcs from a network. Oper. Res. 12(6), 934–940 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Wood, R.K.: Deterministic network interdiction. Math. Comput. Modell. 17(2), 1–18 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Xiong, P., Jirutitijaroen, P., Singh, C.: A distributionally robust optimization model for unit commitment considering uncertain wind power generation. IEEE Trans. Power Syst. 32(1), 39–49 (2017)CrossRefGoogle Scholar
  66. 66.
    Yao, T., Mandala, S.R., Do Chung, B.: Evacuation transportation planning under uncertainty: a robust optimization approach. Netw. Spat. Econ. 9(2), 171 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Zeng, B., Zhao, L.: Solving two-stage robust optimization problems using a column-and-constraint generation method. Oper. Res. Lett. 41(5), 457–461 (2013)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Operations, Business Analytics and Information SystemsUniversity of CincinnatiCincinnatiUSA
  2. 2.Industrial EngineeringClemson UniversityClemsonUSA

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