Advertisement

Mathematical Programming

, Volume 170, Issue 1, pp 43–65 | Cite as

Relaxations and approximations of chance constraints under finite distributions

  • Shabbir Ahmed
  • Weijun Xie
Full Length Paper Series B
  • 460 Downloads

Abstract

Optimization problems with constraints involving stochastic parameters that are required to be satisfied with a prespecified probability threshold arise in numerous applications. Such chance constrained optimization problems involve the dual challenges of stochasticity and nonconvexity. In the setting of a finite distribution of the stochastic parameters, an optimization problem with linear chance constraints can be formulated as a mixed integer linear program (MILP). The natural MILP formulation has a weak relaxation bound and is quite difficult to solve. In this paper, we review some recent results on improving the relaxation bounds and constructing approximate solutions for MILP formulations of chance constraints. We also discuss a recently introduced bicriteria approximation algorithm for covering type chance constrained problems. This algorithm uses a relaxation to construct a solution whose (constraint violation) risk level may be larger than the pre-specified threshold, but is within a constant factor of it, and whose objective value is also within a constant factor of the true optimal value. Finally, we present some new results that improve on the bicriteria approximation factors in the finite scenario setting and shed light on the effect of strong relaxations on the approximation ratios.

Mathematics Subject Classification

90C11 90C15 90C27 

Notes

Acknowledgements

This research has been supported in part by the National Science Foundation Award 1633196 and the Office of Naval Research Grant N00014-18-1-2075.

References

  1. 1.
    Abdi, A., Fukasawa, R.: On the mixing set with a knapsack constraint. Math. Program. 157(1), 191–217 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ahmed, S., Luedtke, J., Song, Y., Xie, W.: Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs. Math. Program. 162(1–2), 51–81 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ahmed, S., Shapiro, A.: Solving chance-constrained stochastic programs via sampling and integer programming. Tutor. Oper. Res. (INFORMS) 10, 261–269 (2008)Google Scholar
  4. 4.
    Alexander, G.J., Baptista, A.M.: A comparison of var and cvar constraints on portfolio selection with the mean-variance model. Manag. Sci. 50(9), 1261–1273 (2004)CrossRefGoogle Scholar
  5. 5.
    Atamtürk, A., Nemhauser, G.L., Savelsbergh, M.W.P.: The mixed vertex packing problem. Math. Program. 89, 35–53 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Beraldi, P., Ruszczyński, A.: A branch and bound method for stochastic integer problems under probabilistic constraints. Optim. Methods Softw. 17(3), 359–382 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bienstock, D., Chertkov, M., Harnett, S.: Chance-constrained optimal power flow: risk-aware network control under uncertainty. SIAM Rev. 56(3), 461–495 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Calafiore, G.C., Campi, M.C.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102, 25–46 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Calafiore, G.C., Campi, M.C.: The scenario approach to robust control design. IEEE Trans. Autom. Control 51(5), 742–753 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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(3), 235–263 (1958)CrossRefGoogle Scholar
  12. 12.
    Deng, Y., Shen, S.: Decomposition algorithm for optimizing multi-server appointment scheduling with chance constraints. Math. Program. 157, 245–276 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dentcheva, D., Prékopa, A., Ruszczynski, A.: Concavity and efficient points of discrete distributions in probabilistic programming. Math. Program. 89, 55–77 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Goyal, V., Ravi, R.: Approximation algorithms for robust covering problems with chance constraints. http://repository.cmu.edu/cgi/viewcontent.cgi?article=1365&context=tepper (2008)
  15. 15.
    Goyal, V., Ravi, R.: A PTAS for the chance-constrained knapsack problem with random item sizes. Oper. Res. Lett. 38(3), 161–164 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Guan, Y., Ahmed, S., Nemhauser, G.L.: Sequential pairing of mixed integer inequalities. Discrete Optim. 4, 21–39 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Günlük, O., Pochet, Y.: Mixing mixed-integer inequalities. Math. Program. 90, 429–457 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132, 31–56 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lejeune, M.A., Margot, F.: Solving chance-constrained optimization problems with stochastic quadratic inequalities. Oper. Res. 64(4), 939–957 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Liu, X., Kılınç-Karzan, F., Küçükyavuz, S.: On intersection of two mixing sets with applications to joint chance-constrained programs. Math. Program. (2018). https://doi.org/10.1007/s10107-018-1231-2
  21. 21.
    Liu, X., Küçükyavuz, S., Luedtke, J.: Decomposition algorithms for two-stage chance-constrained programs. Math. Program. 157(1), 219–243 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Luedtke, J.: A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Math. Program. 146, 219–244 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 122, 247–272 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Miller, A.J., Wolsey, L.A.: Tight formulations for some simple mixed integer programs and convex objective integer programs. Math. Program. 98, 73–88 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nemirovski, A.: On safe tractable approximations of chance constraints. Eur. J. Oper. Res. 219, 707–718 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nemirovski, A., Shapiro, A.: Scenario approximation of chance constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design Under Uncertainty, pp. 3–48. Springer, London (2005)Google Scholar
  28. 28.
    Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pagnoncelli, B.K., Ahmed, S., Shapiro, A.: Sample average approximation method for chance constrained programming: theory and applications. J. Optim. Theory Appl. 142(2), 399–416 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pagnoncelli, B.K., Ahmed, S., Shapiro, A.: Computational study of a chance constrained portfolio selection problem. J. Optim. Theory Appl. 142(2), 399–416 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pavlikov, K., Veremyev, A., Pasiliao, E.L.: Optimization of value-at-risk: computational aspects of mip formulations. J. Oper. Res. Soc. 69, 127–141 (2018)CrossRefGoogle Scholar
  32. 32.
    Pinter, J.: Deterministic approximations of probability inequalities. ZOR Methods Models Oper. Res. 33, 219–239 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Prékopa, A.: Stochastic Programming. Springer, Berlin (1995)CrossRefzbMATHGoogle Scholar
  34. 34.
    Qiu, F., Ahmed, S., Dey, S.S., Wolsey, L.A.: Covering linear programming with violations. INFORMS J. Comput. 26(3), 531–546 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)CrossRefGoogle Scholar
  36. 36.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, vol. 151. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  37. 37.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory, vol. 9. SIAM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  38. 38.
    Shiina, T.: Numerical solution technique for joint chance-constrained programming problem: an application to electric power capacity expansion. J. Oper. Res. Soc. Jpn. 42(2), 128–140 (1999)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Snyder, L.V., Daskin, M.S: Models for reliable supply chain network design. In: Murray, A.T., Grubesic, T.H. (eds.) Critical Infrastructure, pp. 257–289. Springer, Berlin (2007)Google Scholar
  40. 40.
    Song, Y., Luedtke, J., Küçükyavuz, S.: Chance-constrained binary packing problems. INFORMS J. Comput. 26, 735–747 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Swamy, C.: Risk-averse stochastic optimization: probabilistically-constrained models and algorithms for black-box distributions. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 1627–1646. SIAM (2011)Google Scholar
  42. 42.
    Takyi, A.K., Lence, B.J.: Surface water quality management using a multiple-realization chance constraint method. Water Resour. Res. 35(5), 1657–1670 (1999)CrossRefGoogle Scholar
  43. 43.
    Talluri, S., Narasimhan, R., Nair, A.: Vendor performance with supply risk: a chance-constrained dea approach. Int. J. Prod. Econ. 100(2), 212–222 (2006)CrossRefGoogle Scholar
  44. 44.
    van Ackooij, W., Zorgati, R., Henrion, R., Möller, A.: Chance constrained programming and its applications to energy management. In: Dritsas I. (ed.) Stochastic Optimization—Seeing the Optimal for the Uncertain. InTech (2011)Google Scholar
  45. 45.
    Xie, W., Ahmed, S.: Bicriteria approximation of chance constrained covering problems. Submitted for publication, Preprint available at Optimization Online (2018)Google Scholar
  46. 46.
    Xie, W., Ahmed, S.: Distributionally robust chance constrained optimal power flow with renewables: a conic reformulation. IEEE Trans. Power Syst. 33, 1860 (2018). (To appear)CrossRefGoogle Scholar
  47. 47.
    Xie, W., Ahmed, S.: On quantile cuts and their closure for chance constrained optimization problems. Math. Program. (2017).  https://doi.org/10.1007/s10107-017-1190-z
  48. 48.
    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)CrossRefGoogle Scholar
  49. 49.
    Zhao, M., Huang, K., Zeng, B.: A polyhedral study on chance constrained program with random right-hand side. Math. Program. 166(1–2), 19–64 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of Industrial and Systems EngineeringVirginia TechBlacksburgUSA

Personalised recommendations