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

2 CEDRIC - OC - CEDRIC. Optimisation Combinatoire
CEDRIC - Centre d'études et de recherche en informatique et communications
Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01126210
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Submitted on : Friday, March 6, 2015 - 11:45:54 AM
Last modification on : Friday, August 21, 2020 - 1:56:05 PM

### Identifiers

• HAL Id : hal-01126210, version 1

### Citation

Alain Billionnet, Sourour Elloumi, Amélie Lambert. A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation. Journal of Combinatorial Optimization, Springer Verlag, 2014, 28, pp.376-399. ⟨hal-01126210⟩

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