Orbitopal fixing for the full (sub)-orbitope and application to the Unit Commitment Problem

TL;DR

This paper extends orbitopal fixing to the full orbitope, enabling symmetry breaking in binary matrix ILPs, and demonstrates its effectiveness on the Unit Commitment Problem through a new linear-time algorithm and dynamic variant.

Contribution

It introduces a novel extension of orbitopal fixing to the full orbitope, including handling sub-symmetries and a dynamic algorithm for branch-and-bound.

Findings

The extended orbitopal fixing effectively reduces symmetry in ILPs.

The dynamic algorithm improves search efficiency during B&B.

Experimental results outperform state-of-the-art methods on the Unit Commitment Problem.

Abstract

This paper focuses on integer linear programs where solutions are binary matrices, and the corresponding symmetry group is the set of all column permutations. Orbitopal fixing, as introduced by Kaibel et al., is a technique designed to break symmetries in the special case of partitioning (resp. packing) formulations involving matrices with exactly (resp. at most) one 1-entry in each row.The main result of this paper is to extend orbitopal fixing to the full orbitope, defined as the convex hull of binary matrices with lexicographically nonincreasing columns.We determine all the variables whose values are fixed in the intersection of an hypercube face with the full orbitope.Sub-symmetries arising in a given subset of matrices are also considered, thus leading to define the full sub-orbitope in the case of the sub-symmetric group.We propose a linear time orbitopal fixing algorithm handling both symmetries and sub-symmetries.We introduce a dynamic variant of this algorithm where the lexicographical order follows the branching decisions occurring alongthe B&B search. Experimental results for the Unit Commitment Problem are presented. A comparison with state-of-the-art techniques is considered to show the effectiveness of the proposed variants of the algorithm.