Math. Program., Ser. A https://doi.org/10.1007/s10107-018-1306-0 FULL LENGTH PAPER On cutting planes for cardinality-constrained linear programs 1 2 Jinhak Kim · Mohit Tawarmalani · Jean-Philippe P. Richard Received: 12 May 2016 / Accepted: 23 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018 Abstract We derive cutting planes for cardinality-constrained linear programs. These inequalities can be used to separate any basic feasible solution of an LP relaxation of the problem, assuming that this solution violates the cardinality requirement. To derive them, we ﬁrst relax the given simplex tableau into a disjunctive set, expressed in the space of nonbasic variables. We establish that coefﬁcients of valid inequalities for the closed convex hull of this set obey ratios that can be computed directly from the simplex tableau. We show that a transportation problem can be used to separate these inequalities. We then give a constructive procedure to generate violated facet-deﬁning inequalities for the closed convex hull of the disjunctive set using a variant of Prim’s algorithm. Keywords Complementarity/cardinality constraints · Disjunctive sets · Tableau cuts · Equate-and-relax procedure · Concavity cuts · Prim’s algorithm Mathematics Subject Classiﬁcation 90C11 · 52A27 · 90C26 · 90C35 This work
Mathematical Programming – Springer Journals
Published: Jun 6, 2018
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