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Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs

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Abstract

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.

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References

  1. Ahmed, S.: A scenario decomposition algorithm for 0–1 stochastic programs. Oper. Res. Lett. 41, 555–569 (2013)

    Article  MathSciNet  Google Scholar 

  2. Ahmed, S., Papageorgiou, D.J.: Probabilistic set covering with correlations. Oper. Res. 61(2), 438–452 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, X., Sun, J., Sun, X., Zheng, X.: An alternating direction method for chance-constrained optimization problems with discrete distributions. http://www.optimization-online.org/DB_FILE/2012/04/3448.pdf (2012)

  4. Beraldi, P., Ruszczynski, A.: The probabilistic set-covering problem. Oper. Res. 50(6), 956–967 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calafiore, G., Campi, M.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102, 25–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Calafiore, G., Campi, M.: The scenario approach to robust control design. IEEE Trans. Autom. Contr. 51, 742–753 (2006)

    Article  MathSciNet  Google Scholar 

  7. Camm, J.D., Raturi, A.S., Tsubakitani, S.: Cutting big m down to size. Interfaces 20(5), 61–66 (1990)

    Article  Google Scholar 

  8. Carøe, C.C., Schultz, R.: Dual decomposition in stochastic integer programming. Oper. Res. Lett. 24, 37–45 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. 86(3), 595–614 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11, 18–39 (1963)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  12. de Oliveira, W., Sagastizábal, C.: Level bundle methods for oracles with on demand accuracy. Optim. Methods Softw. 29(6), 1180–1209 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. de Oliveira, W., Sagastizábal, C., Lemaréchal, C.: Convex proximal bundle methods in depth: a unified analysis for inexact oracles. Math. Program. 148(1), 241–277 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dentcheva, D.: Optimization Models with Probabilistic Constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design under Uncertainty, pp. 49–97. Springer, London (2006)

    Chapter  Google Scholar 

  15. Dentcheva, D., Martinez, G.: Regularization methods for optimization problems with probabilistic constraints. Math. Program. 138(1), 223–251 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dentcheva, D., Römisch, W.: Duality gaps in nonconvex stochastic optimization. Math. Program. 101, 515–535 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Henrion, R.: A critical note on empirical (sample average, Monte Carlo) approximation of solutions to chance constrained programs. In: Hömberg, D., Tröltzsch, F. (eds.) System Modeling and Optimization. 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, 12–16 September 2011, Revised Selected Papers, pp. 25–37. Springer, Berlin (2013)

  18. Henrion, R., Möller, A.: A gradient formula for linear chance constraints under Gaussian distribution. Math. Oper. Res. 37(3), 475–488 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kiwiel, K.C.: A proximal bundle method with approximate subgradient linearizations. SIAM J. Optim. 16(4), 1007–1023 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132(1), 31–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lejeune, M.A.: Pattern-based modeling and solution of probabilistically constrained optimization problems. Oper. Res. 60(6), 1356–1372 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lemaréchal, C., Renaud, A.: A geometric study of duality gaps, with applications. Math. Program. 90(3), 399–427 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, X., Küçükyavuz, S., Luedtke, J.: Decomposition algorithms for two-stage chance-constrained programs. Math. Program. 157(1), 219–243 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Luedtke, J.: A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Math. Program. 146(1), 219–244 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19, 674–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 12, 247–272 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Magnanti, T., Shapiro, J., Wagner, M.: Generalized linear programming solves the dual. Manag. Sci. 22(11), 1195–1203 (1976)

    Article  MATH  Google Scholar 

  28. Pagnoncelli, B., Ahmed, S., Shapiro, A.: The sample average approximation method for chance constrained programming: theory and applications. J. Optim. Theory Appl. 142, 399–416 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht (1995)

    Book  MATH  Google Scholar 

  30. Qiu, F., Ahmed, S., Dey, S.S., Wolsey, L.A.: Covering linear programming with violations. INFORMS J. Comput. 26(3), 531–546 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rockafellar, R.T., Wets, R.J.-B.: Nonanticipativity and L1-martingales in stochastic optimization problems. In: Wets, R.J.-B. (ed.) Stochastic Systems: Modeling, Identification and Optimization, II, pp. 170–187. Springer, Berlin Heidelberg (1976)

    Chapter  Google Scholar 

  32. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  33. Slater, M.: Lagrange Multipliers Revisited. Technical report, Cowles Foundation for Research in Economics, Yale University (1959)

  34. Song, Y., Luedtke, J.: Branch-and-cut approaches for chance-constrained formulations of reliable network design problems. Math. Program. Comput. 5(4), 397–432 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Song, Y., Luedtke, J., Küçükyavuz, S.: Chance-constrained binary packing problems. INFORMS J. Comput. 26(4), 735–747 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. van Ackooij, W.: Decomposition approaches for block-structured chance-constrained programs with application to hydro-thermal unit commitment. Math. Methods Oper. Res. 80(3), 227–253 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Watson, J.P., Wets, R.J.-B., Woodruff, D.L.: Scalable heuristics for a class of chance-constrained stochastic programs. INFORMS J. Comput. 22, 543–554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang, M., Küçükyavuz, S., Goel, S.: A branch-and-cut method for dynamic decision making under joint chance constraints. Manag. Sci. 60(5), 1317–1333 (2014)

    Article  Google Scholar 

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Acknowledgments

S. Ahmed was supported in part by the National Science Foundation Grant 1331426 and the Office of Naval Research Grant N00014-15-1-2078. J. Luedtke was partly supported by NSF Grant CMMI-0952907 and by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract number DE-AC02-06CH11357. W. Xie was supported in part by the National Science Foundation Grant 1129871 and a fellowship from the Algorithms and Randomness Center at Georgia Tech. The authors acknowledge the valuable comments from the editors and three anonymous reviewers.

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Authors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Shabbir Ahmed & Weijun Xie

  2. Department of Industrial & Systems Engineering and Wisconsin Institute for Discovery, University of Wisconsin-Madison, Madison, WI, USA

    James Luedtke

  3. Virginia Commonwealth University, Richmond, VA, USA

    Yongjia Song

Authors
  1. Shabbir Ahmed
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  2. James Luedtke
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  3. Yongjia Song
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  4. Weijun Xie
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Correspondence to Yongjia Song.

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Ahmed, S., Luedtke, J., Song, Y. et al. Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs. Math. Program. 162, 51–81 (2017). https://doi.org/10.1007/s10107-016-1029-z

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