Approximating two-stage chance-constrained programs with classical probability bounds

  • Bismark SinghEmail author
  • Jean-Paul Watson
Original Paper


We consider a joint-chance constraint (JCC) as a union of sets, and approximate this union using bounds from classical probability theory. When these bounds are used in an optimization model constrained by the JCC, we obtain corresponding upper and lower bounds on the optimal objective function value. We compare the strength of these bounds against each other under two different sampling schemes, and observe that a larger correlation between the uncertainties tends to result in more computationally challenging optimization models. We also observe the same set of inequalities to provide the tightest upper and lower bounds in our computational experiments.


Chance-constrained optimization Bonferroni inequalities Union bounds Stochastic optimization Approximations 



We thank David Morton and Shabbir Ahmed for helpful discussions. This work was supported in part by Sandia’s Laboratory Directed Research and Development (LDRD) program. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. SAND2019-0432 J.


  1. 1.
    Ahmed, S., Shapiro, A.: Solving chance-constrained stochastic programs via sampling and integer programming. Tutor. Oper. Res. 10, 261–269 (2008)Google Scholar
  2. 2.
    Bonferroni, C.: Teoria statistica delle classi e calcolo delle probabilita. Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commericiali di Firenze 8, 3–62 (1936)zbMATHGoogle Scholar
  3. 3.
    Boros, E., Prekopa, A.: Closed form two-sided bounds for probabilities that at least \(r\) and exactly \(r\) out of \(n\) events occur. Math. Oper. Res. 14(2), 317–342 (1989)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brook, A., Kendrick, D., Meeraus, A.: GAMS, a user’s guide (1988)Google Scholar
  5. 5.
    Charnes, A., Cooper, W.W.: Chance-constrained programming. Manag. Sci. 6(1), 73–79 (1959)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11(1), 18–39 (1963)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dawson, D., Sankoff, D.: An inequality for probabilities. Proc. Am. Math. Soc. 18(3), 504–507 (1967)MathSciNetzbMATHGoogle Scholar
  8. 8.
    De Caen, D.: A lower bound on the probability of a union. Discrete Math. 169(1–3), 217–220 (1997)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Galambos, J.: Bonferroni inequalities. Ann. Probab. 5(4), 577–581 (1977)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kuai, H., Alajaji, F., Takahara, G.: A lower bound on the probability of a finite union of events. Discrete Math. 215(1–3), 147–158 (2000)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kwerel, S.M.: Most stringent bounds on aggregated probabilities of partially specified dependent probability systems. J. Am. Stat. Assoc. 70(350), 472–479 (1975)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Liu, X., Küçükyavuz, S., Luedtke, J.: Decomposition algorithms for two-stage chance-constrained programs. Math. Program. 157(1), 219–243 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Luedtke, J.: A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Math. Program. 146(1–2), 219–244 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 122(2), 247–272 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I—convex underestimating problems. Math. Program. 10(1), 147–175 (1976)zbMATHGoogle Scholar
  17. 17.
    Miller, A.J., Wolsey, L.A.: Tight formulations for some simple mixed integer programs and convex objective integer programs. Math. Program. 98(1–3), 73–88 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pintér, J.: Deterministic approximations of probability inequalities. Zeitschrift für Oper. Res. 33(4), 219–239 (1989)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Prékopa, A.: Contributions to the theory of stochastic programming. Math. Program. 4(1), 202–221 (1973)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Prékopa, A.: Boole–Bonferroni inequalities and linear programming. Oper. Res. 36(1), 145–162 (1988)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Prékopa, A.: Sharp bounds on probabilities using linear programming. Oper. Res. 38(2), 227–239 (1990)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Prékopa, A.: The use of discrete moment bounds in probabilistic constrained stochastic programming models. Ann. Oper. Res. 85, 21–38 (1999)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Prékopa, A., Gao, L.: Bounding the probability of the union of events by aggregation and disaggregation in linear programs. Discrete Appl. Math. 145(3), 444–454 (2005)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Sathe, Y., Pradhan, M., Shah, S.: Inequalities for the probability of the occurrence of at least \(m\) out of \(n\) events. J. Appl. Probab. 17(4), 1127–1132 (1980)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Singh, B.: Optimal spatiotemporal resource allocation in public health and renewable energy. Ph.D. thesis, The University of Texas at Austin (2016)Google Scholar
  26. 26.
    Singh, B., Morton, D.P., Santoso, S.: An adaptive model with joint chance constraints for a hybrid wind-conventional generator system. Comput. Manag. Sci. 15, 563–582 (2018)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Yang, J., Alajaji, F., Takahara, G.: Lower bounds on the probability of a finite union of events. SIAM J. Discrete Math. 30(3), 1437–1452 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Discrete Math and OptimizationSandia National LaboratoriesAlbuquerqueUSA
  2. 2.Data Science and Cyber AnalyticsSandia National LaboratoriesLivermoreUSA

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