Detecting and Handling Reflection Symmetries in Mixed-Integer (Nonlinear) Programming

TL;DR

This paper introduces a framework for automatically detecting and handling reflection symmetries in mixed-integer nonlinear programs, enhancing solver performance by addressing broader symmetry classes beyond permutations.

Contribution

It develops a generic detection framework for reflection symmetries in MINLPs and extends existing methods to handle these symmetries, implemented in the SCIP solver.

Findings

Improved symmetry detection in MINLPs.

Enhanced solver efficiency through symmetry handling.

Validated methods with numerical experiments.

Abstract

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.