 
Summary: A Lifted Linear Programming BranchandBound
Algorithm for Mixed Integer Conic Quadratic Programs
Juan Pablo Vielma, Shabbir Ahmed, George L. Nemhauser,
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765
Ferst Drive NW, Atlanta, GA 303320205, USA, {jvielma@isye.gatech.edu, shabbir.ahmed@isye.gatech.edu,
george.nemhauser@isye.gatech.edu}
This paper develops a linear programming based branchandbound algorithm for mixed in
teger conic quadratic programs. The algorithm is based on a higher dimensional or lifted
polyhedral relaxation of conic quadratic constraints introduced by BenTal and Nemirovski.
The algorithm is different from other linear programming based branchandbound algo
rithms for mixed integer nonlinear programs in that, it is not based on cuts from gradient
inequalities and it sometimes branches on integer feasible solutions. The algorithm is tested
on a series of portfolio optimization problems. It is shown that it significantly outperforms
commercial and open source solvers based on both linear and nonlinear relaxations.
Key words: nonlinear integer programming; branch and bound; portfolio optimization
History: February 2007.
1. Introduction
This paper deals with the development of an algorithm for the class of mixed integer non
linear programming (MINLP) problems known as mixed integer conic quadratic program
ming problems. This class of problems arises from adding integrality requirements to conic
