A preprocessing technique for quadratic unconstrained binary optimization
To solve combinatorial optimization problems more easily, it can be valuable to identify a set of necessary or sufficient conditions that an optimal solution of the problem must satisfy. For instance, weak and strong duality conditions in linear programming support the development of the well-known...
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| Veröffentlicht in: | Discrete optimization Jg. 58; S. 100914 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Elsevier B.V
01.11.2025
Elsevier |
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| Abstract | To solve combinatorial optimization problems more easily, it can be valuable to identify a set of necessary or sufficient conditions that an optimal solution of the problem must satisfy. For instance, weak and strong duality conditions in linear programming support the development of the well-known optimization algorithms for these problems. Similarly, the Karush–Kuhn–Tucker conditions give necessary and sufficient conditions for optimality in convex quadratic programming that underlie the development of optimization algorithms in this domain. Although such continuous conditions do not exist for integer programming, some necessary conditions can be derived from Karush–Kuhn–Tucker conditions for the Quadratic Unconstrained Binary Optimization (QUBO) problem. We present these conditions and show how they lead to a derivation of well-known criteria for fixing the values of single variables in the QUBO problem. From this, we show how to generalize these criteria to fix a product of any number of p (integer) literals, which also may be viewed as a generalization of the persistency notion, consisting of clauses of a Constraint Satisfaction Problem that the optimal solution must satisfy. We then couple our list of persistencies with state-of-the-art rules not covered by our approach. The resulting integrated set of conditions for fixing values of variables is tested in computational experiments on instances from standard databases available in the literature, showing that we can fix more variables and reduce problems more fully than previous approaches. |
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| AbstractList | To solve combinatorial optimization problems more easily, it can be valuable to identify a set of necessary or sufficient conditions that an optimal solution of the problem must satisfy. For instance, weak and strong duality conditions in linear programming support the development of the well-known optimization algorithms for these problems. Similarly, the Karush–Kuhn–Tucker conditions give necessary and sufficient conditions for optimality in convex quadratic programming that underlie the development of optimization algorithms in this domain. Although such continuous conditions do not exist for integer programming, some necessary conditions can be derived from Karush–Kuhn–Tucker conditions for the Quadratic Unconstrained Binary Optimization (QUBO) problem. We present these conditions and show how they lead to a derivation of well-known criteria for fixing the values of single variables in the QUBO problem. From this, we show how to generalize these criteria to fix a product of any number of p (integer) literals, which also may be viewed as a generalization of the persistency notion, consisting of clauses of a Constraint Satisfaction Problem that the optimal solution must satisfy. We then couple our list of persistencies with state-of-the-art rules not covered by our approach. The resulting integrated set of conditions for fixing values of variables is tested in computational experiments on instances from standard databases available in the literature, showing that we can fix more variables and reduce problems more fully than previous approaches. |
| ArticleNumber | 100914 |
| Author | Gueye, S. Michelon, P. |
| Author_xml | – sequence: 1 givenname: S. orcidid: 0000-0001-7217-8543 surname: Gueye fullname: Gueye, S. email: serigne.gueye@univ-avignon.fr organization: Avignon Université, Laboratoire d’Informatique d’Avignon, 339 chemin des Meinajariès, Agroparc BP 1228, Avignon Cedex 9, 84911, France – sequence: 2 givenname: P. surname: Michelon fullname: Michelon, P. email: philippe.michelon@univ-avignon.fr organization: Avignon Université, Laboratoire de Mathématiques d’Avignon, 340 chemin des Meinajariès, Agroparc, Avignon, 84140, France |
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| Cites_doi | 10.1016/j.ejor.2017.08.025 10.1057/jors.1990.166 10.1287/mnsc.44.3.336 10.3389/fphy.2014.00005 10.1109/CVPR.2019.00625 10.1007/s12532-023-00236-6 10.1016/S0167-5060(08)70343-1 10.1016/S0166-218X(96)00003-0 |
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| Keywords | Combinatorial optimization 90Cxx Karush–Kuhn–Tucker conditions Constraint programming Quadratic unconstrained binary optimization Persistencies |
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| References | Glover, Lewis, Kochenberger (b6) 2018; 265 Chvátal (b18) 1997; 73 Boros (b7) 2022 (b1) 2022 Hansen (b4) 1979; 5 Hammer, Rudeanu (b3) 1968 Rosenberg (b12) 1972; 6 Rehfeldt, Koch, Shinano (b9) 2023; 15 Glover, Kochenberger, Alidaee (b14) 1998; 44 Beasley (b15) 1998 Glover, Alidaee, Rego, Kochenberger (b16) 2000 Hammer, Hansen (b11) 1981; 26 Beasley (b13) 1990; 41 Ferizovic, Hespe, Lamm, Mnich, Schulz, Strash (b10) 2020 (b17) 2023 Lucas (b2) 2014; 2 Boros, Hammer, Tavares (b5) 2006 J.-H. Lange, B. Andres, P. Swoboda, Combinatorial Persistency Criteria for Multicut and Max-Cut, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, CVPR, 2019. (10.1016/j.disopt.2025.100914_b1) 2022 Hammer (10.1016/j.disopt.2025.100914_b11) 1981; 26 Boros (10.1016/j.disopt.2025.100914_b5) 2006 Ferizovic (10.1016/j.disopt.2025.100914_b10) 2020 Hammer (10.1016/j.disopt.2025.100914_b3) 1968 Glover (10.1016/j.disopt.2025.100914_b16) 2000 Beasley (10.1016/j.disopt.2025.100914_b13) 1990; 41 Rosenberg (10.1016/j.disopt.2025.100914_b12) 1972; 6 (10.1016/j.disopt.2025.100914_b17) 2023 Glover (10.1016/j.disopt.2025.100914_b6) 2018; 265 Boros (10.1016/j.disopt.2025.100914_b7) 2022 Glover (10.1016/j.disopt.2025.100914_b14) 1998; 44 Hansen (10.1016/j.disopt.2025.100914_b4) 1979; 5 Chvátal (10.1016/j.disopt.2025.100914_b18) 1997; 73 Rehfeldt (10.1016/j.disopt.2025.100914_b9) 2023; 15 10.1016/j.disopt.2025.100914_b8 Lucas (10.1016/j.disopt.2025.100914_b2) 2014; 2 Beasley (10.1016/j.disopt.2025.100914_b15) 1998 |
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| SubjectTerms | Combinatorial optimization Computer Science Constraint programming Karush–Kuhn–Tucker conditions Operations Research Persistencies Quadratic unconstrained binary optimization |
| Title | A preprocessing technique for quadratic unconstrained binary optimization |
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