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On 2-stage robust LP with RHS uncertainty: complexity results and applications

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Abstract

We investigate here the class—denoted R-LP-RHSU—of two-stage robust linear programming problems with right-hand-side uncertainty. Such problems arise in many applications e.g: robust PERT scheduling (with uncertain task durations); robust maximum flow (with uncertain arc capacities); robust network capacity expansion problems; robust inventory management; some robust production planning problems in the context of power production/distribution systems. It is shown that such problems can be formulated as large scale linear programs with associated nonconvex separation subproblem. A formal proof of strong NP-hardness for the general case is then provided, and polynomially solvable subclasses are exhibited. Differences with other previously described robust LP problems (featuring row-wise uncertainty instead of column wise uncertainty) are highlighted.

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References

  1. Ahuja R.K., Magnanti R., Orlin J.B.: Network Flows. Prentice Hall, Englewood Cliffs (1993)

    Google Scholar 

  2. Atamtürk A., Zhang M.: Two-stage robust network flow and design under demand uncertainty. Operat. Res. 55(4), 662–673 (2007)

    Article  Google Scholar 

  3. Benders J.F.: Partitioning procedures for solving mixed variables programming problems. Numerische Mathematik 4, 238–252 (1962)

    Article  Google Scholar 

  4. Ben-Ameur W., Kerivin H.: Routing of uncertain traffic demands. Optim. Eng. 3, 283–313 (2005)

    Article  Google Scholar 

  5. Ben Salem, S., Minoux, M.: Gestion de production en contexte incertain, Congrès ROADEF 2009, Nancy (France), 10–12 Février (2009)

  6. Ben-Tal A., Goryashko A., Guslitzer E., Nemirovski A.: Adjustable robust solutions of uncertain linear programs. Math. Programm. 99, 351–376 (2004)

    Article  Google Scholar 

  7. Ben-Tal A., Nemirovski A.: Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998)

    Article  Google Scholar 

  8. Ben-Tal A., Nemirovski A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)

    Article  Google Scholar 

  9. Ben-Tal A., Nemirovski A.: Robust solution of linear programming problems contaminated with uncertain data. Math. Programm. 88, 411–424 (2000)

    Article  Google Scholar 

  10. Ben-Tal A., Nemirovski A.: Robust optimization—methodology and applications. Math. Programm. 92, 453–480 (2002)

    Article  Google Scholar 

  11. Bertsimas D., Sim M.: Robust discrete optimization and network flows. Math. Programm. B 98, 49–71 (2003)

    Article  Google Scholar 

  12. Bertsimas D., Sim M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)

    Article  Google Scholar 

  13. Chekuri C., Oriolo G., Scutella M.G., Shepherd F.B.: Hardness of robust network design. Networks 50(1), 50–54 (2007)

    Article  Google Scholar 

  14. Garey M.R., Johnson DS.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., San Francisco (1979)

    Google Scholar 

  15. Grötschel M., Lovász L., Schrijver A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)

    Article  Google Scholar 

  16. Guslitser, E.: Uncertainty immunized solutions in linear programming, Master’s thesis, Minerva Optimization Center, Technion. http://iew3.technion.ac.il/Labs/Opt/index.php?4 (2002)

  17. Gutiérrez G., Kouvelis P., Kurawarwala A.: A robustness approach to uncapacitated network design problems. EJOR 94(2), 362–376 (1996)

    Article  Google Scholar 

  18. Kouvelis P., Yu G.: Robust Discrete Optimization and its Applications. Kluwer Academic Publishers, Boston (1997)

    Google Scholar 

  19. Laguna M.: Applying robust optimization to capacity expansion of one location in telecommunications with demand uncertainty. Manag. Sci. 44(11), 5101–5110 (1998)

    Article  Google Scholar 

  20. Malcom S., Zenios S.: Robust optimization for power systems capacity expansion under uncertainty. J. ORSA 45(9), 1040–1049 (1994)

    Google Scholar 

  21. Magnanti T., Wong R.: Network design and transportation planning: models and algorithms. Trans. Sci. 18(1), 1–55 (1984)

    Article  Google Scholar 

  22. Minoux M.: Optimum network design models and algorithms in transportation and communication. Int. J. Logist. Res. Appl. 6(1–2), 5–15 (2003)

    Article  Google Scholar 

  23. Minoux M.: Models and algorithms for robust PERT scheduling with time-dependent task durations. Vietnam J. Math. 35(4), 387–398 (2007)

    Google Scholar 

  24. Minoux, M.: Robust linear programming with right-hand-side uncertainty, duality and applications. In: Floudas, L.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn., pp. 3317–3327 (2008)

  25. Minoux M.: Solving some multistage robust decision problems with huge implicitly-defined scenario trees. Algorithm. Oper. Res. 4(1), 1–18 (2008)

    Google Scholar 

  26. Minoux M.: On robust maximum flow with polyhedral uncertainty sets. Optim. Lett. 3, 367–376 (2009)

    Article  Google Scholar 

  27. Minoux M.: Robust network optimization under polyhedral demand uncertainty is NP-hard. Discret. Appl. Math. 158(5), 597–603 (2010)

    Article  Google Scholar 

  28. Minoux, M.: Two-stage robust LP with ellipsoidal RHS uncertainty is NP-hard (submitted)

  29. Ordoñez F., Zhao J.: Robust capacity expansion of network flows. Networks 50(2), 136–145 (2007)

    Article  Google Scholar 

  30. Ouorou, A., Vial, J.P.H.: A model for robust capacity planning for telecommunication networks under demand uncertainty. In: Proceedings 6th International Workshop DRCN 2007, La Rochelle, France

  31. Petrou G., Lemaréchal C., Ouorou A.: An approach to robust network design in telecommunications. RAIRO 41(4), 411–426 (2007)

    Article  Google Scholar 

  32. Soyster A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154–1157 (1973)

    Article  Google Scholar 

  33. Soyster A.L.: Inexact linear programming with generalized resource sets. EJOR 3, 316–321 (1979)

    Article  Google Scholar 

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  1. University Paris-6, Paris, France

    Michel Minoux

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  1. Michel Minoux
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Minoux, M. On 2-stage robust LP with RHS uncertainty: complexity results and applications. J Glob Optim 49, 521–537 (2011). https://doi.org/10.1007/s10898-010-9645-2

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  • DOI: https://doi.org/10.1007/s10898-010-9645-2

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