Defeating symmetry in combinatorial optimization via objective perturbations and hierarchical constraints

Abstract

This article introduces the concept of defeating symmetry in combinatorial optimization via objective perturbations based on, and combined with, symmetry-defeating constraints. Under this novel reformulation, the original objective function is suitably perturbed using a weighted sum of expressions derived from hierarchical symmetry-defeating constraints in a manner that preserves optimality and judiciously guides and curtails the branch-and-bound enumeration process. Computational results are presented for a noise dosage problem, a doubles tennis scheduling problem, and a wagon load-balancing problem to demonstrate the efficacy of using this strategy in concert with traditional hierarchical symmetry-defeating constraints. The proposed methodology is shown to significantly outperform the use of hierarchical constraints or objective perturbations in isolation, as well as the automatic symmetry-defeating feature that is enabled by CPLEX, version 11.2.

Combining ant colony optimization algorithm and dynamic programming technique for solving the covering salesman problem

“…Constraint (9) avoids this situation by enforcing each vertex k to be either covered by itself (y kk ¼ 1) or by the smallest vertex of J 0 k that is also on the tour. For more information on symmetry defeating, we refer the interested reader to Ghoniem and Sherali (2011) and Jans and Desrosiers (2013). Finally, (10) and (11) represent the decision variables.…”

An Integrated Approach for Airline Flight Selection and Timing, Fleet Assignment, and Aircraft Routing

“…By Sherali and Soyster (1983) this can be ensured (within a unit of optimality) by selecting o = 1/f max , wheref max is an upper bound on the maximal value off in (21), and can be taken asf max ≡ s∈S a∈AT F s a m=1 w m n − m + 1 2 noting (2) and (3) and assuming that n ≥ F s a ∀ s ∈ S a ∈ AT . Following Ghoniem and Sherali (2011), we utilize w m ≡ 1/ am b + cm m = 1 M, where a, b, and c, are suitable nonnegative scalars.…”

“…For personal use only, all rights reserved. the set of constraints, we also suitably perturb the original objective function with a weighted sum of terms derived from the corresponding hierarchical constraints as in Ghoniem and Sherali (2011). For this experiment, we considered the following variations: (a) symmetry-breaking constraints alone, versus (b) symmetry-breaking constraints combined with a perturbed objective function, where for the symmetry-breaking constraint (20), we utilize (c) a coefficient of j 2 for each hierarchical term (as stated in (20)), or (d) a coefficient of j for each hierarchical term.…”

“…Furthermore, in light of the objective perturbation strategy suggested by Ghoniem and Sherali (2011) to derive symmetry compatible formulations, we explore perturbing the objective function (1) …”

Using symmetry to optimize over the Sherali–Adams relaxation

“…In the past several years, however, algorithms such as isomorphism pruning [22], orbital branching [28], and orbitopal fixing [17] have been developed which successfully identify and remove equivalent subproblems from the branch-and-bound tree. Other methods include adding symmetry-breaking constraints [31] as well as perturbing the objective function [14]. These symmetry-exploiting algorithms can solve highly-symmetric problems orders of magnitude faster than methods that do not exploit symmetry.…”