Orbital shrinking: Theory and applications

Fischetti, Matteo; Liberti, Leo; Salvagnin, Domenico; Walsh, Toby

doi:10.1016/j.dam.2017.01.015

Cited by 5 publications

(2 citation statements)

References 40 publications

(80 reference statements)

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“…The technique was extended to convex MINLPs and some nonconvex MINLPs having a special structure. A recent survey on the subject is available in [8].…”

Section: Literature Reviewmentioning

confidence: 99%

Exploiting symmetries in mathematical programming via orbital independence

Dias

Liberti

2019

Ann Oper Res

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The presence of symmetries in the solution set of mathematical programs requires the exploration of symmetric subtrees during the execution of Branch-and-Bound type algorithms and yields increases in computation times. When some of the solution symmetries are evident in the formulation, it is possible to deal with symmetries as a preprocessing step. In this sense, implementation-wise, one of the simplest approaches is to break symmetries by adjoining Symmetry-Breaking Constraints (SBCs) to the formulation. Designed to remove some of the symmetric global optima, these constraints are generated from each orbit of the action of the symmetries on the variable index set. Incompatible SBCs however make all of the global optima infeasible. In this paper we introduce and test a new concept of Orbital Independence which we define as independent sets of orbits. We provide necessary conditions for characterizing independent sets of orbits and also prove that such sets embed sufficient conditions to exploit symmetries from two or more distinct orbits concurrently. The theory developed is used to devise an algorithm that potentially identifies the largest independent set of orbits of any mathematical program. Extensive numerical experiments are provided to validate the theoretical results. Keywords combinatorial optimization • symmetry breaking • group theory • quadratic programming This paper is an extension of the work presented in [5].

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“…The technique was extended to convex MINLPs and some nonconvex MINLPs having a special structure. A recent survey on the subject is available in [8].…”

Section: Literature Reviewmentioning

confidence: 99%

Exploiting symmetries in mathematical programming via orbital independence

Dias

Liberti

2019

Ann Oper Res

Self Cite

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“…Another way of exploiting symmetry in B&B is given by isomorphism pruning [30,32,31] and orbital/constrained orbital branching [37,39]. By introducing artificial variables, Fischetti et al [12] reformulate the problem to a reduced problem which considers only variables of symmetry orbits instead of all variables, so-called orbital shrinking.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs

Kouyialis¹,

Wang²,

Misener³

2019

Processes

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Symmetry in mathematical programming may lead to a multiplicity of solutions. In nonconvex optimisation, it can negatively affect the performance of the branch-and-bound algorithm. Symmetry may induce large search trees with multiple equivalent solutions, i.e. with the same optimal value. Dealing with symmetry requires detecting and classifying it first. This work develops methods for detecting groups of symmetry in the formulation of quadratically constrained quadratic optimisation problems via adjacency matrices. Using graph theory, we transform these matrices into Binary Layered Graphs (BLG) and enter them into the software package nauty [36]. Nauty generates important symmetric properties of the original problem.

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An end-to-end software-defined network framework and optimal service development model for SAGN

Lin

Deng

et al. 2022

Telecommun Syst

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Orbital shrinking: Theory and applications

Cited by 5 publications

References 40 publications

Exploiting symmetries in mathematical programming via orbital independence

Exploiting symmetries in mathematical programming via orbital independence

Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs

An end-to-end software-defined network framework and optimal service development model for SAGN

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