Exploiting partial correlations in distributionally robust optimization

  • Divya PadmanabhanEmail author
  • Karthik Natarajan
  • Karthyek Murthy
Full Length Paper Series A


In this paper, we identify partial correlation information structures that allow for simpler reformulations in evaluating the maximum expected value of mixed integer linear programs with random objective coefficients. To this end, assuming only the knowledge of the mean and the covariance matrix entries restricted to block-diagonal patterns, we develop a reduced semidefinite programming formulation, the complexity of solving which is related to characterizing a suitable projection of the convex hull of the set \(\{(\mathbf x , \mathbf x {} \mathbf x '): \mathbf x \in \mathcal {X}\}\) where \(\mathcal {X}\) is the feasible region. In some cases, this lends itself to efficient representations that result in polynomial-time solvable instances, most notably for the distributionally robust appointment scheduling problem with random job durations as well as for computing tight bounds in the newsvendor problem, project evaluation and review technique networks and linear assignment problems. To the best of our knowledge, this is the first example of a distributionally robust optimization formulation for appointment scheduling that permits a tight polynomial-time solvable semidefinite programming reformulation which explicitly captures partially known correlation information between uncertain processing times of the jobs to be scheduled. We also discuss extensions where the random coefficients are assumed to be non-negative and additional overlapping correlation information is available.

Mathematics Subject Classification

90-02 (operations research and mathematical programming. research exposition) 



The research of the first and the second author was partially supported by the MOE Academic Research Fund Tier 2 Grant T2MOE1706, “On the Interplay of Choice, Robustness and Optimization in Transportation” and the SUTD-MIT International Design Center Grant IDG21700101 on “Design of the Last Mile Transportation System: What Does the Customer Really Want?”. The authors would like to thank Teo Chung-Piaw (NUS) for providing some useful references on this research. We would also like to thank the editor Alper Atamturk, the two anonymous reviewers and the associate editor for the detailed perusal of the paper and providing useful suggestions towards improving the paper.


  1. 1.
    Aldous, D.J.: The \(\zeta (2)\) limit in the random assignment problem. Random Struct. Algorithms 18(4), 381–418 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Anstreicher, K.M., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Math. Program. 124, 33–43 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ball, M.O., Colbourn, C.J., Provan, J.S.: Network reliability. In: Handbooks in Operations Research and Management Science, vol. 7, pp. 673–762. Elsevier Science B.V., Amsterdam (1995) Google Scholar
  4. 4.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Cambridge (2009)zbMATHCrossRefGoogle Scholar
  5. 5.
    Berge, C.: Some classes of perfect graphs. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp. 155–166. Academic Press, London (1967) Google Scholar
  6. 6.
    Bertsimas, D., Doan, X.V., Natarajan, K., Teo, C.-P.: Models for minimax stochastic linear optimization problems with risk aversion. Math. Oper. Res. 35(3), 580–602 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bertsimas, D., Natarajan, K., Teo, C.-P.: Probabilistic combinatorial optimization: moments, semidefinite programming, and asymptotic bounds. SIAM J. Optim. 15(1), 185–209 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bertsimas, D., Natarajan, K., Teo, C.-P.: Persistence in discrete optimization under data uncertainty. Math. Program. 108(2), 251–274 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bertsimas, D., Sim, M., Zhang, M.: Adaptive distributionally robust optimization. Manag. Sci. 65(2), 604–618 (2018)CrossRefGoogle Scholar
  11. 11.
    Bomze, I.M., Cheng, J., Dickinson, P.J.C., Lisser, A.: A fresh CP look at mixed-binary QPs: new formulations and relaxations. Math. Program. 166(1–2), 159–184 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bomze, I.M., De Klerk, E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Glob. Optim. 24(2), 163–185 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Buck, M.W., Chan, C.S., Robbins, D.P.: On the expected value of the minimum assignment. Random Struct. Algorithms 21(1), 33–58 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Burer, S.: A gentle, geometric introduction to copositive optimization. Math. Program. 151(1), 89–116 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Cambanis, S., Simons, G., Stout, W.: Inequalities for E k (x, y) when the marginals are fixed. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 36(4), 285–294 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Dickinson, P.J.C.: Geometry of the copositive and completely positive cones. J. Math. Anal. Appl. 380(1), 377–395 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Dickinson, P.J.C., Gijben, L.: On the computational complexity of membership problems for the completely positive cone and its dual. Comput. Optim. Appl. 57(2), 403–415 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Doan, X.V., Natarajan, K.: On the complexity of nonoverlapping multivariate marginal bounds for probabilistic combinatorial optimization problems. Oper. Res. 60(1), 138–149 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Donath, W.E.: Algorithm and average-value bounds for assignment problems. IBM J. Res. Dev. 13(4), 380–386 (1969)zbMATHCrossRefGoogle Scholar
  22. 22.
    Dyer, M.E., Frieze, A.M., Mcdiarmid, C.J.H.: On linear programs with random costs. Math. Program. 35(1), 3–16 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Fulkerson, D.R.: Expected critical path lengths in pert networks. Oper. Res. 10(6), 808–817 (1962)zbMATHCrossRefGoogle Scholar
  24. 24.
    Galichon, A.: Optimal Transport Methods in Economics. Princeton University Press, Princeton (2016)CrossRefGoogle Scholar
  25. 25.
    Goemans, M.X., Kodialam, M.S.: A lower bound on the expected cost of an optimal assignment. Math. Oper. Res. 18(2), 267–274 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Grone, R., Johnson, C.R., Sá, E.M., Wolkowicz, H.: Positive definite completions of partial hermitian matrices. Linear Algebra Appl. 58, 109–124 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Hagstrom, J.N.: Computational complexity of pert problems. Networks 18(2), 139–147 (1988)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hanasusanto, G.A., Kuhn, D.: Conic programming reformulations of two-stage distributionally robust linear programs over wasserstein balls. Oper. Res. 66(3), 849–869 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hanasusanto, G.A., Kuhn, D., Wallace, S.W., Zymler, S.: Distributionally robust multi-item newsvendor problems with multimodal demand distributions. Math. Program. 152(1), 1–32 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Jiang, R., Shen, S., Zhang, Y.: Integer programming approaches for appointment scheduling with random no-shows and service durations. Oper. Res. 65(6), 1638–1656 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Karp, R.M.: An upper bound on the expected cost of an optimal assignment. In: Johnson, D.S., Nishizeki, T., Nozaki, A., Wilf, H.S. (eds.) Discrete Algorithms and Complexity, pp. 1–4. Academic Press, New York (1987)Google Scholar
  32. 32.
    Kong, Q., Lee, C.-Y., Teo, C.-P., Zheng, Z.: Scheduling arrivals to a stochastic service delivery system using copositive cones. Oper. Res. 61(3), 711–726 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Q. 2(1–2), 83–97 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Kurtzberg, J.M.: On approximation methods for the assignment problem. J. ACM 9(4), 419–439 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Laurent, M.: Matrix Completion Problems, pp. 1967–1975. Springer, Boston (2009)Google Scholar
  36. 36.
    Lazarus, A.J.: Certain expected values in the random assignment problem. Oper. Res. Lett. 14(4), 207–214 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Lorentz, G.G.: An inequality for rearrangements. Am. Math. Month. 60(3), 176–179 (1953)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Mak, H.-Y., Rong, Y., Zhang, J.: Appointment scheduling with limited distributional information. Manag. Sci. 61(2), 316–334 (2015)CrossRefGoogle Scholar
  39. 39.
    Mézard, M., Parisi, G.: On the solution of the random link matching problems. J. Phys. 48(9), 1451–1459 (1987)CrossRefGoogle Scholar
  40. 40.
    Möhring, R.H.: Scheduling under uncertainty: bounding the makespan distribution. In: Alt, H. (ed.) Computational Discrete Mathematics, pp. 79–97. Springer, Berlin (2001) CrossRefGoogle Scholar
  41. 41.
    Munkres, J.: Algorithms for the assignment and transportation problems. J. Soc. Ind. Appl. Math. 5(1), 32–38 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Murty, K.G., Kabadi, S.N.: Some np-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117–129 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Natarajan, K., Teo, C.-P.: On reduced semidefinite programs for second order moment bounds with applications. Math. Program. 161(1–2), 487–518 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Natarajan, K., Teo, C.-P., Zheng, Z.: Mixed 0–1 linear programs under objective uncertainty: a completely positive representation. Oper. Res. 59(3), 713–728 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Olin, B.: Asymptotic properties of random assignment problems. PhD thesis, Division of Optimization and Systems Theory, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden (1992)Google Scholar
  46. 46.
    Padberg, M.: The boolean quadric polytope: some characteristics, facets and relatives. Math. Program. 45(1), 139–172 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Parrillo, P.A.: Structured semidefinite programs and semi-algebraic geometry methods in robustness. Ph.D. Thesis, California Institute of Technology (2000)Google Scholar
  48. 48.
    Penrose, R.: A generalized inverse for matrices. Math. Proc. Camb. Philos. Soc. 51(3), 406–413 (1955)zbMATHCrossRefGoogle Scholar
  49. 49.
    Pitowsky, I.: Correlation polytopes: their geometry and complexity. Math. Program. 50(1), 395–414 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Radhakrishna Rao, C., Mitra, S.K.: Generalized inverse of a matrix and its applications. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Theory of Statistics, pp. 601–620. University of California Press, Berkeley (1972)Google Scholar
  51. 51.
    Rose, D.J.: Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32(3), 597–609 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Shogan, A.W.: Bounding distributions for a stochastic pert network. Networks 7(4), 359–381 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Toh, K.C., Todd, M.J., Tutuncu, R.H.: SDPT3—a matlab software package for semidefinite programming. Optim. Methods Softw. 11, 545–581 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Tutuncu, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. Ser. B(95), 189–217 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Van Slyke, R.M.: Monte Carlo methods and the pert problem. Oper. Res. 11(5), 839–860 (1963)CrossRefGoogle Scholar
  56. 56.
    Walkup, D.: On the expected value of a random assignment problem. SIAM J. Comput. 8(3), 440–442 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Wästlund, J.: A proof of a conjecture of buck, chan, and robbins on the expected value of the minimum assignment. Random Struct. Algorithms 26(1–2), 237–251 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Guanglin, X., Burer, S.: A data-driven distributionally robust bound on the expected optimal value of uncertain mixed 0–1 linear programming. Comput. Manag. Sci. 15(1), 111–134 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Boshi, A.K.Y., Burer, S.: Quadratic programs with hollows. Math. Program. 170(2), 541–552 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Zangwill, W.I.: A deterministic multi-period production scheduling model with backlogging. Manag. Sci. 13(1), 105–119 (1966)zbMATHCrossRefGoogle Scholar
  62. 62.
    Zangwill, W.I.: A backlogging model and a multi-echelon model of a dynamic economic lot size production system—a network approach. Manag. Sci. 15(9), 506–527 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Zuluaga, L.F., Peña, J.F.: A conic programming approach to generalized Tchebycheff inequalities. Math. Oper. Res. 30(2), 369–388 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Singapore University of Technology and DesignSingaporeSingapore

Personalised recommendations