Mathematical Programming

, Volume 167, Issue 1, pp 191–234 | Cite as

Data-driven inverse optimization with imperfect information

  • Peyman Mohajerin Esfahani
  • Soroosh Shafieezadeh-Abadeh
  • Grani A. Hanasusanto
  • Daniel KuhnEmail author
Full Length Paper Series B


In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent’s objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent’s true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the predicted decision (i.e., the decision implied by a particular candidate objective) differs from the agent’s actual response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches.

Mathematics Subject Classification

C15 Stochastic programming 90C25 Convex programming 90C47 Minimax problems 



This work was supported by the Swiss National Science Foundation grant BSCGI0_157733.


  1. 1.
    Ackerberg, D., Benkard, C.L., Berry, S., Pakes, A.: Econometric tools for analyzing market outcomes. In: Heckman, J., Leamer, E. (eds.) Handbook of Econometrics, pp. 4171–4276. Elsevier, Amsterdam (2007)Google Scholar
  2. 2.
    Ahmed, S., Guan, Y.: The inverse optimal value problem. Math. Program. A 102, 91–110 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ahuja, R.K., Orlin, J.B.: A faster algorithm for the inverse spanning tree problem. J. Algorithms 34, 177–193 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ahuja, R.K., Orlin, J.B.: Inverse optimization. Oper. Res. 49, 771–783 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Anderson, B.D., Moore, J.B.: Optimal Control: Linear Quadratic Methods. Courier Corporation, North Chelmsford (2007)Google Scholar
  6. 6.
    Aswani, A., Shen, Z.-J.M., Siddiq, A.: Inverse optimization with noisy data. Preprint arXiv:1507.03266 (2015)
  7. 7.
    Bajari, P., Benkard, C.L., Levin, J.: Estimating dynamic models of imperfect competition. Technical Report, National Bureau of Economic Research (2004)Google Scholar
  8. 8.
    Ben-Ayed, O., Blair, C.E.: Computational difficulties of bilevel linear programming. Oper. Res. 38, 556–560 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ben-Tal, A., Ghaoui, El, Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bertsimas, D., Gupta, V., Paschalidis, I.C.: Inverse optimization: a new perspective on the Black–Litterman model. Oper. Res. 60, 1389–1403 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bertsimas, D., Gupta, V., Paschalidis, I.C.: Data-driven estimation in equilibrium using inverse optimization. Math. Program. 153, 595–633 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Boissard, E.: Simple bounds for convergence of empirical and occupation measures in 1-Wasserstein distance. Electron. J. Probab. 16, 2296–2333 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bottasso, C.L., Prilutsky, B.I., Croce, A., Imberti, E., Sartirana, S.: A numerical procedure for inferring from experimental data the optimization cost functions using a multibody model of the neuro-musculoskeletal system. Multibody Syst. Dyn. 16, 123–154 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  15. 15.
    Braess, D., Nagurney, A., Wakolbinger, T.: On a paradox of traffic planning. Transp. Sci. 39, 446–450 (2005)CrossRefGoogle Scholar
  16. 16.
    Bubeck, S.: Convex optimization: algorithms and complexity. Found. Trends Mach. Learn. 8, 231–357 (2015)CrossRefzbMATHGoogle Scholar
  17. 17.
    Burton, D., Toint, P.L.: On an instance of the inverse shortest paths problem. Math. Program. 53, 45–61 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Carr, S., Lovejoy, W.: The inverse newsvendor problem: choosing an optimal demand portfolio for capacitated resources. Manag. Sci. 46, 912–927 (2000)CrossRefzbMATHGoogle Scholar
  19. 19.
    Chan, T.C., Craig, T., Lee, T., Sharpe, M.B.: Generalized inverse multiobjective optimization with application to cancer therapy. Oper. Res. 62, 680–695 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Faragó, A., Szentesi, Á., Szviatovszki, B.: Inverse optimization in high-speed networks. Discrete Appl. Math. 129, 83–98 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fournier, N., Guillin, A.: On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Relat. Fields 162, 707–738 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Friedman, J., Hastie, T., Tibshirani, R.: The Elements of Statistical Learning. Springer, Berlin (2001)zbMATHGoogle Scholar
  23. 23.
    Harker, P., Pang, J.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Heuberger, C.: Inverse combinatorial optimization: a survey on problems, methods, and results. J. Comb. Optim. 8, 329–361 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hochbaum, D.S.: Efficient algorithms for the inverse spanning-tree problem. Oper. Res. 51, 785–797 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Iyengar, G., Kang, W.: Inverse conic programming with applications. Oper. Res. Lett. 33, 319–330 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Keshavarz, A., Wang, Y., Boyd, S.: Imputing a convex objective function. In: IEEE International Symposium on Intelligent Control, pp. 613–619 (2011)Google Scholar
  28. 28.
    Lofberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: IEEE International Symposium on Computer Aided Control Systems Design, pp. 284 –289 (2004)Google Scholar
  29. 29.
    Markowitz, H .M.: Portfolio Selection: Efficient Diversification of Investments. Yale University Press, New Haven (1968)Google Scholar
  30. 30.
    Mohajerin Esfahani, P., Kuhn, D.: Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Math. Program. (2017).
  31. 31.
    Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39, 117–129 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Neumann-Denzau, G., Behrens, J.: Inversion of seismic data using tomographical reconstruction techniques for investigations of laterally inhomogeneous media. Geophys. J. Int. 79, 305–315 (1984)CrossRefGoogle Scholar
  33. 33.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)CrossRefGoogle Scholar
  34. 34.
    Saez-Gallego, J., Morales, J.M., Zugno, M., Madsen, H.: A data-driven bidding model for a cluster of price-responsive consumers of electricity. IEEE Trans. Power Syst. 31, 5001–5011 (2016)CrossRefGoogle Scholar
  35. 35.
    Schaefer, A.J.: Inverse integer programming. Optim. Lett. 3, 483–489 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Shafieezadeh-Abadeh, S., Kuhn, D., Mohajerin Esfahani, P.: Regularization via mass transportation. Preprint arXiv:1710.10016 (2017)
  37. 37.
    Shafieezadeh-Abadeh, S., Esfahani, P.M., Kuhn, D.: Distributionally robust logistic regression. In: Cortes, C., Lawrence, N.D., Lee, D.D., Sugiyama, M., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 28, pp. 1576–1584. Curran Associates, Inc. (2015)Google Scholar
  38. 38.
    Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Terekhov, A.V., Pesin, Y.B., Niu, X., Latash, M.L., Zatsiorsky, V.M.: An analytical approach to the problem of inverse optimization with additive objective functions: an application to human prehension. J. Math. Biol. 61, 423–453 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Troutt, M.D., Pang, W.-K., Hou, S.-H.: Behavioral estimation of mathematical programming objective function coefficients. Manag. Sci. 52, 422–434 (2006)CrossRefzbMATHGoogle Scholar
  41. 41.
    Wang, L.: Cutting plane algorithms for the inverse mixed integer linear programming problem. Oper. Res. Lett. 37, 114–116 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Woodhouse, J.H., Dziewonski, A.M.: Mapping the upper mantle: three-dimensional modeling of earth structure by inversion of seismic waveforms. J. Geophys. Res. 89, 5953–5986 (1984)CrossRefGoogle Scholar
  43. 43.
    Zhang, J., Xu, C.: Inverse optimization for linearly constrained convex separable programming problems. Eur. J. Oper. Res. 200, 671–679 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Delft Center for Systems and ControlTU DelftDelftThe Netherlands
  2. 2.Risk Analytics and Optimization ChairEPFLLausanneSwitzerland
  3. 3.Graduate Program in Operations Research and Industrial EngineeringUT AustinAustinUSA

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