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On the mixing set with a knapsack constraint

Abdi, Ahmad ORCID: 0000-0002-3008-4167 and Fukasawa, Ricardo (2016) On the mixing set with a knapsack constraint. Mathematical Programming, 157 (1). pp. 191-217. ISSN 0025-5610

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Identification Number: 10.1007/s10107-016-0979-5


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.

Item Type: Article
Official URL:
Additional Information: © 2016 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 09 Oct 2019 14:12
Last Modified: 13 Jun 2024 16:45

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