# Approximating two-stage chance-constrained programs with classical probability bounds

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## Abstract

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.

## Keywords

Chance-constrained optimization Bonferroni inequalities Union bounds Stochastic optimization Approximations## Notes

### Acknowledgements

We thank David Morton and Shabbir Ahmed for helpful discussions. This work was supported in part by Sandia’s Laboratory Directed Research and Development (LDRD) program. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. SAND2019-0432 J.

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