Extreme Strong Branching for QCQPs

Abstract: For mixed-integer programs (MIPs), strong branching is a highly effective variable selection method to reduce the number of nodes in the branch-and-bound algorithm. Extending it to nonlinear problems is conceptually simple but practically limited. Branching on a binary variable fixes the variable to 0 or 1, whereas branching on a continuous variable requires an additional decision to choose a branching point. Previous extensions of strong branching predefine this point and then solve $2n$ relaxations where $n$ is the number of candidate variables to branch. We propose extreme strong branching, which evaluates multiple branching points per variable and jointly selects both the branching variable and point based on the objective value improvement. This approach resembles the success of strong branching for MIPs while additionally exploiting bound tightening as a byproduct. For certain types of quadratically constrained quadratic programs (QCQPs), computational experiments show that the extreme strong branching rule outperforms existing commercial solvers.