Skip to main content
SpringerLink
Log in

On reduced semidefinite programs for second order moment bounds with applications

  • Full Length Paper
  • Published:
Mathematical Programming Submit manuscript

Abstract

We show that the complexity of computing the second order moment bound on the expected optimal value of a mixed integer linear program with a random objective coefficient vector is closely related to the complexity of characterizing the convex hull of the points \(\{{1 \atopwithdelims (){\varvec{x}}}{1 \atopwithdelims (){\varvec{x}}}' \ | \ {\varvec{x}} \in {\mathcal {X}}\}\) where \({\mathcal {X}}\) is the feasible region. In fact, we can replace the completely positive programming formulation for the moment bound on \({\mathcal {X}}\), with an associated semidefinite program, provided we have a linear or a semidefinite representation of this convex hull. As an application of the result, we identify a new polynomial time solvable semidefinite relaxation of the distributionally robust multi-item newsvendor problem by exploiting results from the Boolean quadric polytope. For \({\mathcal {X}}\) described explicitly by a finite set of points, our formulation leads to a reduction in the size of the semidefinite program. We illustrate the usefulness of the reduced semidefinite programming bounds in estimating the expected range of random variables with two applications arising in random walks and best–worst choice models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Ahipasaoglu, S.D., Xiaobo, L., Natarajan, K.: A Convex Optimization Approach for Computing Correlated Choice Probabilities with Many Alternatives. Working paper. http://www.optimization-online.org/DB_HTML/2013/09/4034.html (2016)

  2. Albert, A.: Conditions for positive and nonnegative definiteness in terms of pseudoinverses. SIAM J. Appl. Math. 17(2), 434–440 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arnold, B.C., Groeneveld, R.A.: Bounds on expectations of linear systematic statistics based on dependent samples. Math. Oper. Res. 4(4), 441–447 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Anis, A.A., Lloyd, E.H.: On the range of partial sums of a finite number of independent normal variates. Biometrika 40(1–2), 35–42 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  5. Aven, T.: Upper (lower) bounds on the mean of the maximum (minimum) of a number of random variables. J. Appl. Probab. 22, 723–728 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ben-Akiva, M., Lerman, S.R.: Discrete Choice Analysis: Theory and Application to Travel Demand. The MIT Press, Cambridge (1985)

    Google Scholar 

  7. Ben-Tal, A., Ghaoui, L.El, Nemirovskii, A.: Robust Optimization. Princeton University Press, Princeton (2009)

    Book  Google Scholar 

  8. Bertsimas, D., Popescu, I.: On the relation between option and stock prices: an optimization approach. Oper. Res. 50(2), 358–374 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. 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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Boyle, P., Lin, X.S.: Bounds on contingent claims based on several assets. J. Financ. Econ. 46, 383–400 (1997)

    Article  Google Scholar 

  12. Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. Ser. A 120, 479–495 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Carlen, E.: Trace inequalities and quantum entropy: an introductory course. Entropy Quantum 529, 73–140 (2010). (Contemporary Mathematics, Amer. Math. Soc. Providence, RI)

    Article  MATH  MathSciNet  Google Scholar 

  14. Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 596–612 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Deza, M.M., Laurent, M.: Geometry of Cuts and Metrics. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  16. Dickinson, P.J.C.: The Copositive Cone, the Completely Positive Cone and their Generalisations. PhD Thesis (2013)

  17. Dickinson, P.J.C., Gibjen, L.: On the computational complexity of membership problems for the completely positive cone and its dual. Comput. Optim. Appl. 57(2), 403–415 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. van Dijk, B., Fok, D., Paap, R.: A rank-ordered logit model with unobserved heterogeneity in ranking capabilities. J. Appl. Econom. 27(5), 831–846 (2012)

    Article  MathSciNet  Google Scholar 

  19. Feller, W.: The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Stat. 22, 427–432 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  20. Greene, W.H., Hensher, D.A.: Modeling Ordered Choices: A Primer. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  21. Gumbel, E.J.: The maximum of the mean largest value and of the range. Ann. Math. Stat. 25, 76–84 (1954)

    Article  MATH  Google Scholar 

  22. Hanasusanto, G.A., Kuhn, D., Wallace, S.W., Zymler, S.: Distributionally robust multi-item newsvendor problems with multimodal demand distributions. Math. Program. 152, 1–32 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hartley, H.O., David, H.A.: Universal bounds for mean range and extreme observations. Ann. Math. Stat. 25(1), 85–99 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hurst, H.E.: Long term storage capacity of reservoirs. Trans. Am. Soc. Civil Eng. 116, 770–799 (1950)

    Google Scholar 

  25. Isii, K.: On the sharpness of Chebyshev-type inequalities. Ann. Math. Stat. 14, 185–197 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  26. Karlin, S., Studden, W.J.: Tchebycheff Systems: With Applications in Analysis and Statistics. Wiley, New York (1966)

    MATH  Google Scholar 

  27. Lasserre, J.B.: Bounds on measures satisfying moment conditions. Ann. Appl. Probab. 12(3), 1114–1137 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lefevre, C.: Bounds on the expectations of linear combinations of order statistics with applications to PERT networks. Stoch. Anal. Appl. 4, 351–356 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  29. Li, X., Karthik, N., Teo, C.P., Zheng, Z.: Distributionally robust mixed integer linear programs: persistency models with applications. Eur. J. Oper. Res. 233(3), 459–473 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  30. Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)

  31. Luce, R.D., Suppes, P.: Preference, utility and subjective probability. In: Luce, R.D., Bush, R.R., Galanter, E. (eds.) Handbook of Mathematical Psychology, vol. 3, pp. 235–406. Wiley, New York (1965)

    Google Scholar 

  32. Marley, A.A.J., Louviere, J.J.: Some probabilistic models of best, worst and best–worst choices. J. Math. Psychol. 49, 464–480 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  33. Marshall, A.W., Olkin, I.: A one-sided inequality of the Chebyshev type. Ann. Math. Stat. 31(2), 488–491 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  34. Marshall, A.W., Olkin, I.: Multivariate Chebyshev inequalities. Ann. Math. Stat. 31(4), 1001–1014 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  35. McFadden, D.: Conditional logit analysis of qualitative choice behavior. In: Zarembka, P. (ed.) Front. Econom., pp. 105–142. Academic Press, New York (1974)

    Google Scholar 

  36. Mesa, O.J., Poveda, G.: The Hurst Effect: the scale of fluctuation approach. Water Resour. Res. 29(12), 3995–4002 (1993)

    Article  Google Scholar 

  37. Mishra, V.K., Natarajan, K., Tao, H., Teo, C.P.: Choice modeling with semidefinite optimization when utilities are correlated. IEEE Trans. Autom. Control 57(10), 2450–2463 (2012)

    Article  Google Scholar 

  38. Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117–129 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  39. Nagaraja, H.N.: Some finite sample results for the selection differential. Ann. Inst. Stat. Math. 33, 437–448 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  40. Natarajan, K., Teo, C.P., Zheng, Z.: Mixed zero-one linear programs under objective uncertainty: a completely positive representation. Oper. Res. 59(3), 713–728 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  41. Nesterov, Y.: Squared functional systems and optimization problems. In: Frenk, H., et al. (eds.) High Performance Optimization, pp. 405–440. Kluwer Academic Publishers, Dordrecht (2000)

    Chapter  Google Scholar 

  42. Olkin, I., Pratt, J.W.: A multivariate Tchebycheff inequality. Ann. Math. Stat. 29(1), 226–234 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  43. Padberg, M.: The boolean quadric polytope: some characteristics, facets and relatives. Math. Program. 45, 139–172 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  44. Palma, A., de Kilani, K., Laffond, G.: Best and worst choices. In: Working Paper (2013)

  45. Papdatos, N.: Expectation bounds on linear estimators from dependent samples. J. Stat. Plan. Inference 93, 17–28 (2001)

    Article  MathSciNet  Google Scholar 

  46. Plackett, R.L.: Limits of the ratio of mean range to standard deviation. Biometrika 34(1–2), 120–122 (1947)

    Article  MATH  MathSciNet  Google Scholar 

  47. Penrose, R.: A generalized inverse for matrices. Proc. Camb. Philos. Soc. 51, 406–413 (1955)

    Article  MATH  Google Scholar 

  48. Popescu, I.: Robust mean-covariance solutions for stochastic optimization and applications. Oper. Res. 55(1), 98–112 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  49. Rao, C.R., Mishra, S.K.: Generalized inverse of a matrix and its applications. Proc. Camb. Philos. Soc. 103, 269–276 (1988)

    Article  Google Scholar 

  50. Shapiro, A.: Extremal problems on the set of nonnegative definite matrices. Linear Algebra Appl. 67, 7–18 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  51. Salce, L., Zanardo, P.: Completely positive matrices and positivity of least squares solutions. Linear Algebra Appl. 178, 201–216 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  52. Toh, K.C., Todd, M.J., Tutuncu, R.H.: SDPT3—a Matlab software package for semidefinite programming. Optim. Methods Softw. 11, 545–581 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  53. Tutuncu, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. Ser. B 95, 189–217 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  54. Vandenberghe, L., Boyd, S., Comanor, K.: Generalized Chebyshev bounds via semidefinite programming. SIAM Rev. 49, 52–64 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  55. Van Ophem, H., van Stam, P., Praag, B.: Multichoice logit: modeling incomplete preference rankings of classical concerts. J. Bus. Econ. Stat. 17, 117–128 (1999)

    Article  Google Scholar 

  56. Yu, Y.L., Li,Y., Schuurmans, D., Szepesvári, C.: Ageneral projection property for distribution families. In: Bengio, Y., Schuurmans, D., Lafferty, J.D., Williams, C.K.I., Culotta, A. (eds.) Advances in Neural Information Processing Systems, pp. 2232–2240. Curran Associates, Inc., Vancouver, British Columbia (2009)

  57. Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program. 137(1–2), 167–198 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank two anonymous referees, the Associate Editor and the editor Alexander Shapiro for their very useful comments on the paper. The authors would also like to thank A. A. Marley and Zheng Zhichao for useful discussions on this topic.

Author information

Authors and Affiliations

  1. Engineering System and Design, Singapore University of Technology and Design, Singapore, 138682, Singapore

    Karthik Natarajan

  2. Department of Decision Sciences, NUS Business School, Singapore, 117591, Singapore

    Chung-Piaw Teo

Authors
  1. Karthik Natarajan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chung-Piaw Teo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chung-Piaw Teo.

Additional information

The research of the first author was partly supported by the Grant IDG31300105 on ‘Optimization for Complex Discrete Choice’ funded by the SUTD-MIT International Design Center and the MOE Tier 2 Grant Number MOE2013-T2-2-168 on Distributional Robust Optimization for Consumer Choice in Transportation Systems.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 259 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Natarajan, K., Teo, CP. On reduced semidefinite programs for second order moment bounds with applications. Math. Program. 161, 487–518 (2017). https://doi.org/10.1007/s10107-016-1019-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-016-1019-1

Keywords

Search

Navigation