Simple odd β-cycle inequalities for binary polynomial optimization
We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd β -cycle inequalities valid for this polytope, showed that these generally have Chvátal rank 2 with respect to the standard relaxation and that, together...
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| Veröffentlicht in: | Mathematical programming Jg. 206; H. 1-2; S. 203 - 238 |
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| Abstract | We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd
β
-cycle inequalities valid for this polytope, showed that these generally have Chvátal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd
β
-cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chvátal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs. Finally, we report about computational results of our prototype implementation. The simple odd
β
-cycle inequalities sometimes help to close more of the integrality gap in the experiments; however, the preliminary implementation has substantial computational cost, suggesting room for improvement in the separation algorithm. |
|---|---|
| AbstractList | We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd
$$\beta $$
β
-cycle inequalities valid for this polytope, showed that these generally have Chvátal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd
$$\beta $$
β
-cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chvátal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs. Finally, we report about computational results of our prototype implementation. The simple odd
$$\beta $$
β
-cycle inequalities sometimes help to close more of the integrality gap in the experiments; however, the preliminary implementation has substantial computational cost, suggesting room for improvement in the separation algorithm. We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd β -cycle inequalities valid for this polytope, showed that these generally have Chvátal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd β -cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chvátal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs. Finally, we report about computational results of our prototype implementation. The simple odd β -cycle inequalities sometimes help to close more of the integrality gap in the experiments; however, the preliminary implementation has substantial computational cost, suggesting room for improvement in the separation algorithm. |
| Author | Del Pia, Alberto Walter, Matthias |
| Author_xml | – sequence: 1 givenname: Alberto orcidid: 0000-0001-8428-3914 surname: Del Pia fullname: Del Pia, Alberto organization: Department of Industrial and Systems Engineering and Wisconsin Institute for Discovery, University of Wisconsin-Madison – sequence: 2 givenname: Matthias orcidid: 0000-0002-6615-5983 surname: Walter fullname: Walter, Matthias email: m.walter@utwente.nl organization: Department of Applied Mathematics, University of Twente |
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| Cites_doi | 10.1016/j.disopt.2017.02.001 10.1287/opre.36.3.493 10.1007/BF02592023 10.1007/BF01386390 10.1287/ijoo.2019.0049 10.1007/s12532-019-00169-z 10.1287/moor.2016.0804 10.1287/moor.2021.1121 10.1007/s10898-020-00972-2 10.1137/16M1095998 10.1088/0305-4470/31/16/004 10.1137/050646500 10.1145/28869.28874 10.1287/opre.22.1.180 10.1007/s10878-009-9264-3 10.1051/jphys:01987004804055900 10.1007/BF02592196 10.1007/s10288-018-0392-4 10.1007/978-3-031-06901-7_14 10.1088/1742-5468/2010/05/L05003 10.1137/1.9781611977073.105 |
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| Keywords | Cutting planes Binary polynomial optimization Separation algorithm |
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| Snippet | We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd
β
-cycle... We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd $$\beta $$ β... |
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| SubjectTerms | Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical |
| Title | Simple odd β-cycle inequalities for binary polynomial optimization |
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