  In this paper we consider the use of extended formulations in LP-based
algorithms for mixed integer conic quadratic programming (MICQP). Extended
formulations have been used by Vielma, Ahmed and Nemhauser (2008) and Hijazi,
Bonami and Ouorou (2013) to construct algorithms for MICQP that can provide a
significant computational advantage. The first approach is based on an extended
or lifted polyhedral relaxation of the Lorentz cone by Ben-Tal and Nemirovski
(2001) that is extremely economical, but whose approximation quality cannot be
iteratively improved. The second is based on a lifted polyhedral relaxation of
the euclidean ball that can be constructed using techniques introduced by
Tawarmalani and Sahinidis (2005). This relaxation is less economical, but its
approximation quality can be iteratively improved. Unfortunately, while the
approach of Vielma, Ahmed and Nemhauser is applicable for general MICQP
problems, the approach of Hijazi, Bonami and Ouorou can only be used for MICQP
problems with convex quadratic constraints. In this paper we show how a
homogenization procedure can be combined with the technique by Tawarmalani and
Sahinidis to adapt the extended formulation used by Hijazi, Bonami and Ouorou
to a class of conic mixed integer programming problems that include general
MICQP problems. We then compare the effectiveness of this new extended
formulation against traditional and extended formulation-based algorithms for
MICQP. We find that this new formulation can be used to improve various
LP-based algorithms. In particular, the formulation provides an
easy-to-implement procedure that, in our benchmarks, significantly improved the
performance of commercial MICQP solvers.
