An algorithm for binary linear chance-constrained problems using IIS

  • Gianpiero CanessaEmail author
  • Julian A. Gallego
  • Lewis Ntaimo
  • Bernardo K. Pagnoncelli


We propose an algorithm based on infeasible irreducible subsystems to solve binary linear chance-constrained problems with random technology matrix. By leveraging on the problem structure we are able to generate good quality upper bounds to the optimal value early in the algorithm, and the discrete domain is used to guide us efficiently in the search of solutions. We apply our methodology to individual and joint binary linear chance-constrained problems, demonstrating the ability of our approach to solve those problems. Extensive numerical experiments show that, in some cases, the number of nodes explored by our algorithm is drastically reduced when compared to a commercial solver.


Chance-constrained programming Infeasible irreducible subsystems Integer programming 



  1. 1.
    Abdelaziz, F.B., Aouni, B., El Fayedh, R.: Multi-objective stochastic programming for portfolio selection. Eur. J. Oper. Res. 177(3), 1811–1823 (2007)zbMATHGoogle Scholar
  2. 2.
    Abdi, A., Fukasawa, R.: On the mixing set with a knapsack constraint. Math. Program. 157(1), 191–217 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ahmed, S., Papageorgiou, D.J.: Probabilistic set covering with correlations. Oper. Res. 61(2), 438–452 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beasley, J.E.: An algorithm for set covering problem. Eur. J. Oper. Res. 31(1), 85–93 (1987)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Beraldi, P., Ruszczyński, A.: The probabilistic set-covering problem. Oper. Res. 50(6), 956–967 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Campi, M.C., Garatti, S.: The exact feasibility of randomized solutions of uncertain convex programs. SIAM J. Optimiz. 19(3), 1211–1230 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Campi, M.C., Garatti, S., Prandini, M.: The scenario approach for systems and control design. Annu. Rev. Control 33(2), 149–157 (2009)Google Scholar
  8. 8.
    Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4(3), 235–263 (1958)Google Scholar
  9. 9.
    Chinneck, J.W.: Feasibility and infeasibility in optimization: algorithms and computational methods, vol. 118. Springer, Berlin (2007)zbMATHGoogle Scholar
  10. 10.
    Dentcheva, D., Prékopa, A., Ruszczynski, A.: Concavity and efficient points of discrete distributions in probabilistic programming. Math. Program. 89(1), 55–77 (2000)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gleeson, J., Ryan, J.: Identifying minimally infeasible subsystems of inequalities. ORSA J. Comput. 2(1), 61–63 (1990)zbMATHGoogle Scholar
  12. 12.
    Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132(1–2), 31–56 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kumral, M.: Application of chance-constrained programming based on multi-objective simulated annealing to solve a mineral blending problem. Eng. Optimiz. 35(6), 661–673 (2003)MathSciNetGoogle Scholar
  14. 14.
    Lejeune, M., Noyan, N.: Mathematical programming approaches for generating p-efficient points. Eur. J. Oper. Res. 207(2), 590–600 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lejeune, M.A.: Pattern-based modeling and solution of probabilistically constrained optimization problems. Oper. Res. 60(6), 1356–1372 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optimiz. 19(2), 674–699 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pagnoncelli, B., Ahmed, S., Shapiro, A.: Sample average approximation method for chance constrained programming: theory and applications. J. Optimiz. Theory Appl. 142(2), 399–416 (2009)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Prékopa, A.: Dual method for the solution of a one-stage stochastic programming problem with random rhs obeying a discrete probability distribution. Z. Oper. Res. 34(6), 441–461 (1990)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Saxena, A., Goyal, V., Lejeune, M.A.: Mip reformulations of the probabilistic set covering problem. Math. Program. 121(1), 1–31 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Song, Y., Luedtke, J.R., Küçükyavuz, S.: Chance-constrained binary packing problems. INFORMS J. Comput. 26(4), 735–747 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Tanner, M.W., Ntaimo, L.: IIS branch-and-cut for joint chance-constrained stochastic programs and application to optimal vaccine allocation. Eur. J. Oper. Res. 207(1), 290–296 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Zhu, M., Taylor, D.B., Sarin, S.C., Kramer, R., et al.: Chance constrained programming models for risk-based economic and policy analysis of soil conservation. Agric. Resour. Econ. Rev. 23(1), 58–65 (1994)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universidad Adolfo IbañezSantiagoChile
  2. 2.AT Kearney IncChicagoUSA
  3. 3.Texas A&M UniversityCollege StationUSA
  4. 4.Universidad Adolfo IbañezSantiagoChile
  5. 5.IEMS Department, Northwestern UniversityEvanstonUSA

Personalised recommendations