Extreme Strong Branching for QCQPs

TL;DR

This paper introduces extreme strong branching for QCQPs, evaluating multiple branching points per variable to improve variable selection and outperform existing solvers in certain cases.

Contribution

It extends strong branching to nonlinear problems by jointly selecting variables and branching points, enhancing solution efficiency for QCQPs.

Findings

Outperforms existing commercial solvers on certain QCQPs

Evaluates multiple branching points per variable

Exploits bound tightening as a byproduct

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.