A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation

Let be a general mixed-integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. Our approach to solve is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables , additional quadratic c...

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Vydáno v:	Journal of combinatorial optimization Ročník 28; číslo 2; s. 376 - 399
Hlavní autoři:	Billionnet, Alain, Elloumi, Sourour, Lambert, Amélie
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Boston Springer US 01.08.2014 Springer Verlag
Témata:	Combinatorics Computational Complexity Computer Science Convex and Discrete Geometry Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation Quadratic convex relaxation General mixed-integer quadratic programming Experiments Branch and Bound Branche et lié aux relaxation convexe quadratique experiments Générale mixte-entier quadratique expériences quadratic convex relaxation
ISSN:	1382-6905, 1573-2886
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Abstract	Let be a general mixed-integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. Our approach to solve is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables , additional quadratic constraints , a convex objective function, and a set of valid inequalities. Contrarily to the reformulation proposed in Billionnet et al. (Math Program 131(1):381–401, 2012 ), the equivalent problem cannot be directly solved by a standard solver. Here, we propose a new Branch and Bound process based on the relaxation of the non-convex constraints to solve . Computational experiences are carried out on pure- and mixed-integer quadratic instances. The results show that the solution time of most of the considered instances with up to 60 variables is improved by our Branch and Bound algorithm in comparison with the approach of Billionnet et al. ( 2012 ) and with the general mixed-integer nonlinear solver BARON (Sahinidis and Tawarmalani, Global optimization of mixed-integer nonlinear programs, user’s manual, 2010 ).
AbstractList	Let be a general mixed-integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. Our approach to solve is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables , additional quadratic constraints , a convex objective function, and a set of valid inequalities. Contrarily to the reformulation proposed in Billionnet et al. (Math Program 131(1):381–401, 2012 ), the equivalent problem cannot be directly solved by a standard solver. Here, we propose a new Branch and Bound process based on the relaxation of the non-convex constraints to solve . Computational experiences are carried out on pure- and mixed-integer quadratic instances. The results show that the solution time of most of the considered instances with up to 60 variables is improved by our Branch and Bound algorithm in comparison with the approach of Billionnet et al. ( 2012 ) and with the general mixed-integer nonlinear solver BARON (Sahinidis and Tawarmalani, Global optimization of mixed-integer nonlinear programs, user’s manual, 2010 ). Let (MQP) be a general mixed-integer quadratic program that consists of minimizing a quadratic function f(x) = x^TQx +c^Tx subject to linear constraints. Our approach to solve (MQP) is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables y_{ij}, additional quadratic constraints y_{ij}=x_ix_j, a convex objective function, and a set of valid inequalities. Contrarily to the reformulation proposed in MIQCR, the equivalent problem cannot be directly solved by a standard solver. Here, we propose a new Branch and Bound process based on the relaxation of the non-convex constraints y_{ij}=x_ix_j to solve $(MQP)$. Computational experiences are carried out on pure- and mixed-integer quadratic instances. The results show that the solution time of most of the considered instances with up to 60 variables is improved by our Branch and Bound algorithm in comparison with MIQCR and with the general mixed-integer nonlinear solver BARON.
Author	Elloumi, Sourour Lambert, Amélie Billionnet, Alain
Author_xml	– sequence: 1 givenname: Alain surname: Billionnet fullname: Billionnet, Alain organization: CEDRIC-ENSIIE – sequence: 2 givenname: Sourour surname: Elloumi fullname: Elloumi, Sourour organization: CEDRIC-ENSIIE – sequence: 3 givenname: Amélie surname: Lambert fullname: Lambert, Amélie email: amelie.lambert@cnam.fr organization: CEDRIC-CNAM
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Keywords	Quadratic convex relaxation General mixed-integer quadratic programming Experiments Branch and Bound Branche et lié aux relaxation convexe quadratique experiments Générale mixte-entier quadratique expériences quadratic convex relaxation
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Snippet	Let be a general mixed-integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. Our approach to solve is first... Let (MQP) be a general mixed-integer quadratic program that consists of minimizing a quadratic function f(x) = x^TQx +c^Tx subject to linear constraints. Our...
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SubjectTerms	Combinatorics Computational Complexity Computer Science Convex and Discrete Geometry Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation
Title	A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation
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