A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed-Integer Conic Quadratic Programs

This paper develops a linear-programming-based branch-and-bound algorithm for mixed-integer conic quadratic programs. The algorithm is based on a known higher-dimensional or lifted polyhedral relaxation of conic quadratic constraints. The algorithm is different from other linear-programming-based branch-and-bound algorithms 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.