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Chance constrained 0–1 quadratic programs using copulas

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Abstract

In this paper, we study 0–1 quadratic programs with joint probabilistic constraints. The row vectors of the constraint matrix are assumed to be normally distributed but are not supposed to be independent. We propose a mixed integer linear reformulation and provide an efficient semidefinite relaxation of the original problem. The dependence of the random vectors is handled by the means of copulas. Finally, numerical experiments are conducted to show the strength of our approach.

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References

  1. Beasley, J.B.: OR-Library (2012). http://people.brunel.ac.uk/mastjjb/jeb/info.html

  2. Boyd, S.P., Vandenberghe, L.: Applications of semidefinite programming. App. Numer. Math. 29, 283–299 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambfidge (2004). Seventh printing with corrections, 2009

    Book  MATH  Google Scholar 

  4. Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009). doi:10.1007/s10107-008-0223-z

    Article  MATH  MathSciNet  Google Scholar 

  5. Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manage. Sci. 4(3), 235–263 (1958)

    Article  Google Scholar 

  6. Cheng, J., Gicquel, C., Lisser, A.: A second-order cone programming approximation to joint chance-constrained linear programs. In: Mahjoub A., Markakis V., Milis I., Paschos V. (eds.) Combinatorial Optimization, Lecture Notes in Computer Science, vol. 7422, pp. 71–80. Springer, Berlin/Heidelberg (2012). doi:10.1007/978-3-642-32147-4_8

  7. Cheng, J., Lisser, A.: A second-order cone programming approach for linear programs with joint probabilistic constraints. Operat. Res. Lett. 40(5), 325–328 (2012). doi:10.1016/j.orl.2012.06.008

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheng, J., Lisser, A.: A completely positive representation of 0–1 linear programs with joint probabilistic constraints. Operat. Res. Lett. 41(6), 597–601 (2013). doi:10.1016/j.orl.2013.08.008

    Article  MATH  MathSciNet  Google Scholar 

  9. Dentcheva, D.: Optimization models with probabilistic constraints. In: Shapiro, A., Dentcheva, D., Ruszczyński, A. (eds.) Lectures on Stochastic Programming: Modeling and Theory, MOS-SIAM Series on Optimization, vol. 9, pp. 87–153. SIAM, Philadelphia (2009)

    Chapter  Google Scholar 

  10. Fortet, R.: Applications de l’algebre de boole en recherche operationelle. Revue Francaise de Recherche Operationelle 4, 17–26 (1960)

  11. Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. Henrion, R., Strugarek, C.: Convexity of chance constraints with dependent random variables: the use of copulae. In: Bertocchi M., Consigli G., Dempster M.A.H. (eds.) Stochastic Optimization Methods in Finance and Energy, International Series in Operations Research and Management Science, vol. 163, pp. 427–439. Springer, New York (2011). doi:10.1007/978-1-4419-9586-5_17

  13. IBM: IBM ILOG CPLEX Optimizer. http://www.ibm.com/software/integration/optimization/cplex/

  14. László, L.: On the shannon capacity of a graph. IEEE Trans. Inf. Theory 25(1), 1–7 (1979)

    Article  MATH  Google Scholar 

  15. Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  16. Parillo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. thesis, California Institute of Technology (2000)

  17. Prékopa, A.: Stochastic Programming. Akadémiai Kiadó, Budapest (1995)

    Book  Google Scholar 

  18. Prékopa, A.: Probabilistic programming. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10, pp. 267–352. Elsevier, Amsterdam (2003)

    Google Scholar 

  19. Singh V. Jain S., T.A.: Risk and reliability analysis: a handbook for civil and environmental engineers, vol. 1. ASCE (2007)

  20. Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)

    Article  MathSciNet  Google Scholar 

  21. Thakur, L.S.: Error analysis for convex separable programs: the piecewise linear approximation and the bounds on the optimal objective value. SIAM J. Appl. Math. 34(4), 704–714 (1978). doi:10.1137/0134059

    Article  MATH  MathSciNet  Google Scholar 

  22. Zhang, L.: S.V.: Gumbel-Hougaard copula for trivariate rainfall frequency analysis. J. Hydrol. Eng. 12(4), 409–419 (2007)

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Acknowledgments

This research was supported by Fondation Mathématiques Jacques Hadamard, PGMO/IROE grant No. 2012-042H.

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Authors and Affiliations

  1. Laboratoire de Recherche en Informatique, Université Paris Sud XI, Bât. 650, 91405, Orsay Cedex, France

    Jianqiang Cheng, Michal Houda & Abdel Lisser

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  1. Jianqiang Cheng
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  2. Michal Houda
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  3. Abdel Lisser
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Correspondence to Abdel Lisser.

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Cheng, J., Houda, M. & Lisser, A. Chance constrained 0–1 quadratic programs using copulas. Optim Lett 9, 1283–1295 (2015). https://doi.org/10.1007/s11590-015-0854-y

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  • DOI: https://doi.org/10.1007/s11590-015-0854-y

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