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Optimization Letters

, Volume 13, Issue 2, pp 281–294 | Cite as

Inverse optimization for multi-objective linear programming

  • Mostafa NaghaviEmail author
  • Ali Asghar Foroughi
  • Masoud Zarepisheh
Original paper
  • 35 Downloads

Abstract

This paper generalizes inverse optimization for multi-objective linear programming where we are looking for the least problem modifications to make a given feasible solution a weak efficient solution. This is a natural extension of inverse optimization for single-objective linear programming with regular “optimality” replaced by the “Pareto optimality”. This extension, however, leads to a non-convex optimization problem. We prove some special characteristics of the problem, allowing us to solve the non-convex problem by solving a series of convex problems.

Keywords

Multi-objective linear programming Linear programming Inverse optimization Efficiency 

Notes

Acknowledgements

This work was partially supported by the MSK Cancer Center Support Grant/Core Grant (P30 CA008748).

References

  1. 1.
    Ahuja, R.K., Orlin, J.B.: Inverse optimization. Oper. Res. 49(5), 771–783 (2001).  https://doi.org/10.1287/opre.49.5.771.10607 MathSciNetzbMATHGoogle Scholar
  2. 2.
    Akbari, Z., Peyghami, M.R.: An interior-point algorithm for solving inverse linear optimization problem. Optimization 61(4), 373–386 (2012).  https://doi.org/10.1080/02331934.2011.637111 MathSciNetzbMATHGoogle Scholar
  3. 3.
    Annetts, J.E., Audsley, E.: Multiple objective linear programming for environmental farm planning. J. Oper. Res. Soc. 53(9), 933–943 (2002).  https://doi.org/10.1057/palgrave.jors.2601404 zbMATHGoogle Scholar
  4. 4.
    Bazaraa, M.S., Jarvis, J.J., Sherali, H.D.: Linear Programming and Network Flows. Wiley, London (2011)zbMATHGoogle Scholar
  5. 5.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  6. 6.
    Burton, D., Toint, P.L.: On an instance of the inverse shortest paths problem. Math. Program. 53(1), 45–61 (1992).  https://doi.org/10.1007/BF01585693 MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chan, T.C.Y., Craig, T., Lee, T., Sharpe, M.B.: Generalized inverse multiobjective optimization with application to cancer therapy. Oper. Res. 62(3), 680–695 (2014).  https://doi.org/10.1287/opre.2014.1267 MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chan, T.C.Y., Lee, T.: Trade-off preservation in inverse multi-objective convex optimization. Eur. J. Oper. Res. 270(1), 25–39 (2018).  https://doi.org/10.1016/j.ejor.2018.02.045 MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chan, T.C.Y., Lee, T., Terekhov, D.: Inverse optimization: closed-form solutions, geometry, and goodness of fit. Manag. Sci. pp. 1 – 21 (2018).  https://doi.org/10.1287/mnsc.2017.2992
  10. 10.
    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2006)zbMATHGoogle Scholar
  11. 11.
    Ghate, A.: Inverse optimization in countably infinite linear programs. Oper. Res. Lett. 43(3), 231–235 (2015).  https://doi.org/10.1016/j.orl.2015.02.004 MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hamacher, H.W., Küfer, K.H.: Inverse radiation therapy planning: a multiple objective optimization approach. Discrete Appl. Math. 118(1), 145–161 (2002).  https://doi.org/10.1016/S0166-218X(01)00261-X MathSciNetzbMATHGoogle Scholar
  13. 13.
    Heuberger, C.: Inverse combinatorial optimization: a survey on problems, methods, and results. J. Comb. Optim. 8(3), 329–361 (2004).  https://doi.org/10.1023/B:JOCO.0000038914.26975.9b MathSciNetzbMATHGoogle Scholar
  14. 14.
    Iyengar, G., Kang, W.: Inverse conic programming with applications. Oper. Res. Lett. 33(3), 319–330 (2005).  https://doi.org/10.1016/j.orl.2004.04.007 MathSciNetzbMATHGoogle Scholar
  15. 15.
    Luc, D.T.: Multiobjective Linear Programming. Springer, Berlin (2016)zbMATHGoogle Scholar
  16. 16.
    Mostafaee, A., Hladík, M., Černỳ, M.: Inverse linear programming with interval coefficients. J. Comput. Appl. Math. 292, 591–608 (2016).  https://doi.org/10.1016/j.cam.2015.07.034 MathSciNetzbMATHGoogle Scholar
  17. 17.
    Murty, K.G.: Linear Programming. Wiley, London (1983)zbMATHGoogle Scholar
  18. 18.
    Roland, J., Smet, Y.D., Figueira, J.R.: Inverse multi-objective combinatorial optimization. Discrete Appl. Math. 161(16), 2764–2771 (2013).  https://doi.org/10.1016/j.dam.2013.04.024 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Schaefer, A.J.: Inverse integer programming. Optim. Lett. 3(4), 483–489 (2009).  https://doi.org/10.1007/s11590-009-0131-z MathSciNetzbMATHGoogle Scholar
  20. 20.
    Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation, and Applications. Wiley, London (1986)zbMATHGoogle Scholar
  21. 21.
    Steuer, R.E., Na, P.: Multiple criteria decision making combined with finance: a categorized bibliographic study. Eur. J. Oper. Res. 150(3), 496–515 (2003).  https://doi.org/10.1016/S0377-2217(02)00774-9 zbMATHGoogle Scholar
  22. 22.
    Zhang, J., Liu, Z.: Calculating some inverse linear programming problems. J. Comput. Appl. Math. 72(2), 261–273 (1996).  https://doi.org/10.1016/0377-0427(95)00277-4 MathSciNetzbMATHGoogle Scholar
  23. 23.
    Zhang, J., Liu, Z.: A further study on inverse linear programming problems. J. Comput. Appl. Math. 106(2), 345–359 (1999).  https://doi.org/10.1016/S0377-0427(99)00080-1 MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of QomQomIran
  2. 2.Department of Medical PhysicsMemorial Sloan Kettering Cancer CenterNew YorkUSA

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