  Symmetries in mixed-integer (nonlinear) programs (MINLP), if not handled
appropriately, are known to negatively impact the performance of (spatial)
branch-and-bound algorithms. Usually one thus tries to remove symmetries from
the problem formulation or is relying on a solver that automatically detects
and handles symmetries. While modelers of a problem can handle various kinds of
symmetries, automatic symmetry detection and handling is mostly restricted to
permutation symmetries. This article therefore develops techniques such that
also black-box solvers can automatically detect and handle a broader class of
symmetries.
  Inspired from geometric packing problems such as the kissing number problem,
we focus on reflection symmetries of MINLPs. We develop a generic and easily
applicable framework that allows to automatically detect reflection symmetries
for MINLPs. To handle this broader class of symmetries, we discuss
generalizations of state-of-the-art methods for permutation symmetries, and
develop dedicated symmetry handling methods for special reflection symmetry
groups. Our symmetry detection framework has been implemented in the
open-source solver SCIP and we provide a comprehensive discussion of the
implementation. The article concludes with a detailed numerical evaluation of
our symmetry handling methods when solving MINLPs.
