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**Sample approximation technique for mixed-integer stochastic programming problems with several chance constraints.**
*(English)*
Zbl 1245.90073

Summary: The paper deals with sample approximation applied to stochastic programming problems with chance constraints. We extend results on rates of convergence for problems with mixed-integer bounded sets of feasible solutions and several chance constraints. We derive estimates on the sample size necessary to get a feasible solution of the original problem using sample approximation. We present an application to a vehicle routing problem with time windows, random travel times, and random demand.

### MSC:

90C15 | Stochastic programming |

90C11 | Mixed integer programming |

90C59 | Approximation methods and heuristics in mathematical programming |

### Keywords:

chance constrained problems; sample approximation; rate of convergence; sample size; stochastic vehicle routing problem
Full Text:
DOI

### References:

[1] | M. Branda, Nonconvex stochastic programming problems: formulations, sample approximations and stability. Ph.D. Thesis, Faculty of Mathematics and Physics, Charles University in Prague, 2010. |

[2] | M. Branda, Stochastic programming problems with generalized integrated chance constraints, Accepted to Optimization, doi:10.1080/02331934.2011.587007. · Zbl 1252.90056 |

[3] | Branda, M.; Dupačová, J., Approximations and contamination bounds for probabilistic programs, Annals of operations research, 193, 3-19, (2012), See also SPEPS 2008, 13 · Zbl 1254.90133 |

[4] | Calafiore, G.; Campi, M.C., Uncertain convex programs: randomized solutions and confidence levels, Mathematical programming, series A, 102, 25-46, (2008) · Zbl 1177.90317 |

[5] | Charnes, A.; Cooper, W.W.; Symonds, G.H., Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil, Management science, 4, 235-263, (1958) |

[6] | J. Dupačová, M. Kopa, Robustness in stochastic programs with risk constraints, Accepted to Annals of Operations Research. doi:10.1007/s10479-010-0824-9. |

[7] | Hoeffding, W., Probability inequalities for sums of bounded random variables, Journal of the American statistical association, 58, 301, 13-30, (1963) · Zbl 0127.10602 |

[8] | Kaňková, V., On the convergence rate of empirical estimates in chance constrained stochastic programming, Kybernetika, 26, 6, 449-461, (1990) · Zbl 0721.90057 |

[9] | Kenyon, A.S.; Morton, D.P., Stochastic vehicle routing with random travel times, Transportation science, 37, 1, 69-82, (2003) |

[10] | Li, X.; Tian, P.; Leung, S.C.H., Vehicle routing problems with time windows and stochastic travel and service times: models and algorithm, International journal of production economics, 125, 137-145, (2010) |

[11] | Luedtke, J.; Ahmed, S., A sample approximation approach for optimization with probabilistic constraints, SIAM journal on optimization, 19, 674-699, (2008) · Zbl 1177.90301 |

[12] | Pagnoncelli, B.; Ahmed, S.; Shapiro, A., Computational study of a chance constrained portfolio selection problem, Optimization online, (2008) |

[13] | Pagnoncelli, B.; Ahmed, S.; Shapiro, A., Sample average approximation method for chance constrained programming: theory and applications, Journal optimization theory and applications, 142, 399-416, (2009) · Zbl 1175.90306 |

[14] | Prékopa, A., Dual method for the solution of a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution, Mathematical methods of operations research, 34, 441-461, (1990) · Zbl 0724.90048 |

[15] | Prékopa, A., Stochastic programming, (1995), Kluwer Dordrecht and Académiai Kiadó, Budapest · Zbl 0834.90098 |

[16] | Prékopa, A., Probabilistic programming, (), 267-352 |

[17] | Shapiro, A., Monte Carlo sampling methods, (), 353-426 |

[18] | Shen, Z.; Ordonez, F.; Dessouky, M.M., The stochastic vehicle routing problem for minimum unmet demand, optimization and logistics challenges in the enterprise, Springer optimization and its applications, 30, 349-371, (2009) · Zbl 1172.90342 |

[19] | Wang, W.; Ahmed, S., Sample average approximation of expected value constrained stochastic programs, Operations research letters, 36, 515-519, (2008) · Zbl 1210.90131 |

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