Stochastic programming problems with generalized integrated chance constraints. (English) Zbl 1252.90056

Summary: If the constraints in an optimization problem are dependent on a random parameter, we would like to ensure that they are fulfilled with a high level of reliability. The most natural way is to employ chance constraints. However, the resulting problem is very hard to solve. We propose an alternative formulation of stochastic programs using penalty functions. The expectations of penalties can be left as constraints leading to generalized integrated chance constraints, or incorporated into the objective as a penalty term. We show that the penalty problems are asymptotically equivalent under quite mild conditions. We discuss applications of sample-approximation techniques to the problems with generalized integrated chance constraints and propose rates of convergence for the set of feasible solutions. We will direct our attention to the case when the set of feasible solutions is finite, which can appear in integer programming. The results are then extended to the bounded sets with continuous variables. Additional binary variables are necessary to solve sample-approximated chance-constrained problems, leading to a large mixed-integer non-linear program. On the other hand, the problems with penalties can be solved without adding binary variables; just continuous variables are necessary to model the penalties. The introduced approaches are applied to the blending problem leading to comparably reliable solutions.


90C15 Stochastic programming
Full Text: DOI


[1] Ahmed S, Tutorials in Operations Research pp 261– (2008)
[2] Bazara MS, Nonlinear Programming: Theory and Algorithms (1993)
[3] Branda M, Stochastic Programming E-Print Series (SPEPS), 2010
[4] Branda M, Proceedings of Mathematical Methods in Economics pp 67– (2010)
[5] Branda M, Ann. Oper. Res.
[6] DOI: 10.1016/0377-2217(91)90333-Q · Zbl 0726.90048
[7] Dupačová J, Ann. Oper. Res.
[8] DOI: 10.1023/A:1019244405392 · Zbl 0990.90084
[9] Fügenschuh A, Handbook on Discrete Optimization, Handbooks in Operations Research and Management Science 12 pp 69– (2005)
[10] DOI: 10.1007/978-3-642-01689-9 · Zbl 1204.62088
[11] Klein Haneveld WK, Lecture, Notes in Economics and Mathematical Systems 274 (1986)
[12] DOI: 10.1007/s10287-005-0007-3 · Zbl 1136.90424
[13] DOI: 10.1137/070702928 · Zbl 1177.90301
[14] DOI: 10.1007/s10107-008-0247-4 · Zbl 1184.90115
[15] Nocedal J, Numerical Optimization, 2. ed. (2006)
[16] Pagnoncelli B, Optimization (online) (2008)
[17] DOI: 10.1007/s10957-009-9523-6 · Zbl 1175.90306
[18] DOI: 10.1007/BF01421551 · Zbl 0724.90048
[19] Prékopa A, Stochastic Programming (1995)
[20] Prékopa A, Stochastic Programming, Handbook in Operations Research and Management Science 10 pp 483– (2003)
[21] DOI: 10.1287/mnsc.16.11.708 · Zbl 0211.22601
[22] Shapiro A, Stochastic Programming, Handbook in Operations Research and Management Science 10 pp 483– (2003)
[23] DOI: 10.1016/j.orl.2008.05.003 · Zbl 1210.90131
[24] E. Žampachová, and M. Mrázek,Stochastic optimization in beam design and its reliability check, inMENDEL 2010 – 16th International Conference on Soft Computing, R. Matousek, ed., Mendel Journal series, FME BUT, Brno, 2010, pp. 405–410
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.