Orbitopal fixing for the full (sub-)orbitope and application to 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 in [12], 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 along the 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. DefinitionsThroughout the paper, we consider an Integer Linear Program (ILP) of the form min c(x) | x ∈ X , with X ⊆ P(m, n) and c : P(m, n) → Rwhere P(m, n) is the set of m × n binary matrices. A symmetry is defined as a permutation π of the columns {1, ..., n} such that for any solution matrix x ∈ X , matrix π(x) is also solution and has same cost, i.e., π(x) ∈ X and c(x) = c(π(x)). The symmetry group G of ILP (1) is the set of all * pascale.bendotti@edf.fr † pierre.fouilhoux@lip6.fr ‡ cecile.rottner@edf.fr such permutations. Symmetry group G partitions the solution set X into orbits, i.e., two matrices are in the same orbit if there exists a permutation in G sending one to the other. Symmetries arising in ILP can impair the solution process, in particular when symmetric solutions lead to an excessively large branch and bound (B&B) search tree (see survey [19]). Symmetry detection techniques are proposed in [15,3]. Various techniques, so called symmetry-breaking techniques, are available to handle symmetries in ILP of the form (1). The general idea is, in each orbit, to pick one solution, defined as the representative, and then restrict the solution set to the set of all representatives.A technique is said to be full-symmetry-breaking (resp. partial-symmetry-breaking) if the solution set is exactly (resp. partially) restricted to the representative set. Moreover, such a technique may introduce some specific branching rules that interfere with the B&B search. This can forbid exploiting a user-defined branching rule or, even, the default solver branching settings. A symmetrybreaking technique is said to be flexible if at any node of the B&B tree, the branching rule can be derived from any linear inequality on the variables. Such a technique can be based on specific...

Sub-Symmetry-Breaking Inequalities for ILP with Structured Symmetry

“…Some symmetry-breaking techniques operate during the branching process. In particular, branching techniques (modified orbital branching [31]) or fixing techniques (orbitopal fixing for the full orbitope [5]) have been applied to the UCP. These are typically implemented using a callback instruction which deeply affects the performance of commercial solvers like Cplex.…”

Large-scale unit commitment under uncertainty: an updated literature survey

“…Moreover, severe symmetry issues [378] must be faced [323,380], as these can significantly degrade the performances of the B&B approach. Recent symmetry breaking techniques, known as orbital branching (a terminology pinned down in [377]) or orbitopal fixing (e.g., [268]) are applied to UC in [60,375]. All these difficulties, not shared by UC with DC or AC network constraints, require a nontrivial extension of the "classic" MILP UC models.…”

Symmetry-breaking inequalities for ILP with structured sub-symmetry

“…A subproblem is problem pILP q restricted to a subset of X . In [5], symmetries arising in solution subsets of pILP q are called sub-symmetries. Such sub-symmetries may not exist in G.…”

“…Most techniques based on branching and pruning rules [28,33,13] are either full symmetrybreaking or flexible. Variable fixing [19,5] is both full symmetry-breaking and flexible. Other symmetry-breaking techniques rely on full or partial symmetry-breaking inequalities.…”

“…The convex hull of all mˆn binary matrices with lexicographically non-increasing columns is called the full orbitope [20]. Sub-symmetries and sub-orbitopes are introduced in [5] as a generalization of symmetries and full orbitopes to a given set of matrix subsets. The aim in this paper is to develop a general framework, that enables deriving sub-symmetry-breaking inequalities designed to handle simultaneously symmetries and sub-symmetries in symmetric groups.…”