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We describe strong convex valid inequalities for conic quadratic mixed 0–1 optimization. These inequalities can be utilized for solving numerous practical nonlinear discrete optimization problems from value-at-risk minimization to queueing system design, from robust interdiction to assortment optimization through appropriate conic quadratic mixed 0–1 relaxations. The inequalities exploit the submodularity of the binary restrictions and are based on the polymatroid inequalities over binaries for the diagonal case. We prove that the convex inequalities completely describe the convex hull of a single conic quadratic constraint as well as the rotated cone constraint over binary variables and unbounded continuous variables. We then generalize and strengthen the inequalities by incorporating additional constraints of the optimization problem. Computational experiments on mean-risk optimization with correlations, assortment optimization, and robust conic quadratic optimization indicate that the new inequalities strengthen the convex relaxations substantially and lead to significant performance improvements.
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Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraints
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 $f(a^Tx)$, 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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- Award ID(s):
- 2007814
- PAR ID:
- 10522712
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Mathematical Programming
- Volume:
- 201
- Issue:
- 1-2
- ISSN:
- 0025-5610
- Page Range / eLocation ID:
- 803 to 861
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10107-022-01921-5
