Analyzing the numerical correctness of branch-and-bound decisions for mixed-integer programming

TL;DR

This paper investigates the numerical correctness of branch-and-bound decisions in mixed-integer programming solvers, verifying their validity in exact arithmetic and analyzing the impact of numerical errors on solver decisions.

Contribution

It introduces an a posteriori verification method for branch-and-bound decisions in MIP solvers, providing the first such analysis of their correctness in practice.

Findings

Most decisions are correct in typical cases.

Numerical errors, when they occur, affect only a small number of nodes.

Errors are more likely on numerically challenging instances.

Abstract

Most state-of-the-art branch-and-bound solvers for mixed-integer linear programming rely on limited-precision floating-point arithmetic and use numerical tolerances when reasoning about feasibility and optimality during their search. While the practical success of floating-point MIP solvers bears witness to their overall numerical robustness, it is well-known that numerically challenging input can lead them to produce incorrect results. Even when their final answer is correct, one critical question remains: Were the individual decisions taken during branch-and-bound justified, i.e., can they be verified in exact arithmetic? In this paper, we attempt a first such a posteriori analysis of a pure LP-based branch-and-bound solver by checking all intermediate decisions critical to the correctness of the result: accepting solutions as integer feasible, declaring the LP relaxation infeasible, and pruning subtrees as subopti mal. Our computational study in the academic MIP solver SCIP confirms the expectation that in the overwhelming majority of cases, all decisions are correct. When errors do occur on numerically challenging instances, they typically affect only a small, typically single-digit, amount of leaf nodes that would require further processing.