Approximating Two-Stage Chance-Constrained Programs with Classical Probability Bounds

Singh, Bismark; Watson, Jean-Paul

doi:10.1007/s11590-019-01387-z

Title: Approximating Two-Stage Chance-Constrained Programs with Classical Probability Bounds

Journal Article · Tue Jan 01 00:00:00 EST 2019 · Optimization Letters

DOI:https://doi.org/10.1007/s11590-019-01387-z· OSTI ID:1492794

Singh, Bismark ^[1]; Watson, Jean-Paul ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Discrete Math & Optimization
Sandia National Lab. (SNL-CA), Livermore, CA (United States). Data Science & Cyber Analytics

We consider a joint-chance constraint (JCC) as a union of sets, and approximate this union using bounds from classical probability theory. When these bounds are used in an optimization model constrained by the JCC, we obtain corresponding upper and lower bounds on the optimal objective function value. We compare the strength of these bounds against each other under two different sampling schemes, and observe that a larger correlation between the uncertainties tends to result in more computationally challenging optimization models. We also observe the same set of inequalities to provide the tightest upper and lower bounds in our computational experiments.

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Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1492794

Report Number(s):: SAND-2019-0432J; 671570

Journal Information:: Optimization Letters, Journal Name: Optimization Letters; ISSN 1862-4472

Publisher:: Springer NatureCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 2 works

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References (23)

A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support Luedtke, James Mathematical Programming, Vol. 146, Issue 1-2 https://doi.org/10.1007/s10107-013-0684-6	journal	May 2013
An adaptive model with joint chance constraints for a hybrid wind-conventional generator system Singh, Bismark; Morton, David P.; Santoso, Surya Computational Management Science, Vol. 15, Issue 3-4 https://doi.org/10.1007/s10287-018-0309-x	journal	April 2018
Most Stringent Bounds on Aggregated Probabilities of Partially Specified Dependent Probability Systems Kwerel, Seymour M. Journal of the American Statistical Association, Vol. 70, Issue 350 https://doi.org/10.1080/01621459.1975.10479893	journal	June 1975
Contributions to the theory of stochastic programming Prékopa, András Mathematical Programming, Vol. 4, Issue 1 https://doi.org/10.1007/BF01584661	journal	December 1973
Boole-Bonferroni Inequalities and Linear Programming Prékopa, András Operations Research, Vol. 36, Issue 1 https://doi.org/10.1287/opre.36.1.145	journal	February 1988
Deterministic Equivalents for Optimizing and Satisficing under Chance Constraints Charnes, A.; Cooper, W. W. Operations Research, Vol. 11, Issue 1 https://doi.org/10.1287/opre.11.1.18	journal	February 1963
Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems McCormick, Garth P. Mathematical Programming, Vol. 10, Issue 1 https://doi.org/10.1007/BF01580665	journal	December 1976
Closed Form Two-Sided Bounds for Probabilities that At Least r and Exactly r Out of n Events Occur Boros, Endre; Prékopa, András Mathematics of Operations Research, Vol. 14, Issue 2 https://doi.org/10.1287/moor.14.2.317	journal	May 1989
A lower bound on the probability of a union de Caen, D. Discrete Mathematics, Vol. 169, Issue 1-3 https://doi.org/10.1016/S0012-365X(96)00107-0	journal	May 1997
An integer programming approach for linear programs with probabilistic constraints Luedtke, James; Ahmed, Shabbir; Nemhauser, George L. Mathematical Programming, Vol. 122, Issue 2 https://doi.org/10.1007/s10107-008-0247-4	journal	October 2008
Chance-Constrained Programming Charnes, A.; Cooper, W. W. Management Science, Vol. 6, Issue 1 https://doi.org/10.1287/mnsc.6.1.73	journal	October 1959
Lower Bounds on the Probability of a Finite Union of Events Yang, Jun; Alajaji, Fady; Takahara, Glen SIAM Journal on Discrete Mathematics, Vol. 30, Issue 3 https://doi.org/10.1137/15M100866X	journal	January 2016
A Sample Approximation Approach for Optimization with Probabilistic Constraints Luedtke, James; Ahmed, Shabbir SIAM Journal on Optimization, Vol. 19, Issue 2 https://doi.org/10.1137/070702928	journal	January 2008
A lower bound on the probability of a finite union of events Kuai, H.; Alajaji, F.; Takahara, G. Discrete Mathematics, Vol. 215, Issue 1-3 https://doi.org/10.1016/S0012-365X(99)00246-0	journal	March 2000
Bounding the probability of the union of events by aggregation and disaggregation in linear programs Prékopa, András; Gao, Linchun Discrete Applied Mathematics, Vol. 145, Issue 3 https://doi.org/10.1016/j.dam.2004.03.003	journal	January 2005
An inequality for probabilities Dawson, D. A.; Sankoff, D. Proceedings of the American Mathematical Society, Vol. 18, Issue 3 https://doi.org/10.1090/S0002-9939-1967-0211424-0	journal	March 1967
Bonferroni Inequalities Galambos, Janos The Annals of Probability, Vol. 5, Issue 4 https://doi.org/10.1214/aop/1176995765	journal	August 1977
Sharp Bounds on Probabilities Using Linear Programming Prékopa, András Operations Research, Vol. 38, Issue 2 https://doi.org/10.1287/opre.38.2.227	journal	April 1990
Tight formulations for some simple mixed integer programs and convex objective integer programs Miller, Andrew J.; Wolsey, Laurence A. Mathematical Programming, Vol. 98, Issue 1-3 https://doi.org/10.1007/s10107-003-0397-3	journal	September 2003
Decomposition algorithms for two-stage chance-constrained programs Liu, Xiao; Küçükyavuz, Simge; Luedtke, James Mathematical Programming, Vol. 157, Issue 1 https://doi.org/10.1007/s10107-014-0832-7	journal	October 2014
Inequalities for the probability of the occurrence of at least m out of n events Sathe, Y. S.; Pradhan, Meena; Shah, S. P. Journal of Applied Probability, Vol. 17, Issue 4 https://doi.org/10.2307/3213226	journal	December 1980
GAMS, a user's guide Brook, Anthony; Kendrick, David; Meeraus, Alexander ACM SIGNUM Newsletter, Vol. 23, Issue 3-4 https://doi.org/10.1145/58859.58863	journal	December 1988
Inequalities for the probability of the occurrence of at least m out of n events Sathe, Y. S.; Pradhan, Meena; Shah, S. P. Journal of Applied Probability, Vol. 17, Issue 04 https://doi.org/10.1017/s002190020009745x	journal	December 1980

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