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On the mixing set with a knapsack constraint

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Abstract

We study a substructure appearing in mixed-integer programming reformulations of chance-constrained programs with stochastic right-hand-sides over a finite discrete distribution, which we call the mixing set with a knapsack constraint. Recently, Luedtke et al. (Math. Program. 122(2):247–272, 2010) and Küçükyavuz (Math Program 132(1):31–56, 2012) studied valid inequalities for such sets. However, most of their results were focused on the equal probabilities case (when the knapsack constraint reduces to a cardinality constraint). In this paper, we focus on the general probabilities case (general knapsack constraint). We characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the results previously derived for the equal probabilities case. Our results allow for a deep understanding of the relationship that the set under consideration has with the knapsack polytope. Moreover, they allow us to establish benchmarks that can be used to identify when a relaxation will be useful for the considered types of reformulations of chance-constrained programs.

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Notes

  1. Sometimes the finite discrete distribution given is an approximation of an unknown continuous distribution, obtained via a Sample Average Approximation (SAA) technique.

  2. Prior to our work, the main focus seems to have been on the equal probabilities case, justified by employing the SAA technique. Our more general framework of the general probabilities case allows for more sophisticated techniques such as Importance Sampling. See the recent work of Barrera et al. [6] for further details.

  3. As shown in Luedtke [13], some valid inequalities for Q are also of value in the extremely general case where each mixing constraint is replaced by a set of arbitrary linear constraints.

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Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

    Ahmad Abdi & Ricardo Fukasawa

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  1. Ahmad Abdi
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  2. Ricardo Fukasawa
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Correspondence to Ricardo Fukasawa.

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Partially supported by NSERC Discovery Grant RGPIN-05623.

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Abdi, A., Fukasawa, R. On the mixing set with a knapsack constraint. Math. Program. 157, 191–217 (2016). https://doi.org/10.1007/s10107-016-0979-5

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  • DOI: https://doi.org/10.1007/s10107-016-0979-5

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