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Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraints

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Abstract

We study the polyhedral convex hull structure of a mixed-integer set which arises in a class of cardinality-constrained concave submodular minimization problems. This class of problems has an objective function in the form of \(f(a^\top x)\), where f is a univariate concave function, a is a non-negative vector, and x is a binary vector of appropriate dimension. Such minimization problems frequently appear in applications that involve risk-aversion or economies of scale. We propose three classes of strong valid linear inequalities for this convex hull and specify their facet conditions when a has two distinct values. We show how to use these inequalities to obtain valid inequalities for general a that contains multiple values. We further provide a complete linear convex hull description for this mixed-integer set when a contains two distinct values and the cardinality constraint upper bound is two. Our computational experiments on the mean-risk optimization problem demonstrate the effectiveness of the proposed inequalities in a branch-and-cut framework.

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Acknowledgements

We thank the editor and the reviewers for the helpful comments that improved this paper. In particular, we thank the reviewer for providing the example in Remark 2. This research is supported, in part, by NSF grant 2007814 and ONR grant N00014-22-1-2602. This research is also supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University, which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.

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  1. Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA

    Qimeng Yu & Simge Küçükyavuz

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  1. Qimeng Yu
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Correspondence to Simge Küçükyavuz.

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Yu, Q., Küçükyavuz, S. Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraints. Math. Program. 201, 803–861 (2023). https://doi.org/10.1007/s10107-022-01921-5

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