On cutting planes for cardinality-constrained linear programs

On cutting planes for cardinality-constrained linear programs 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Programming Springer Journals

On cutting planes for cardinality-constrained linear programs

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematics of Computing; Numerical Analysis; Combinatorics; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics
ISSN
0025-5610
eISSN
1436-4646
D.O.I.
10.1007/s10107-018-1306-0
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Mathematical ProgrammingSpringer Journals

Published: Jun 6, 2018

References

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