Compact Mathematical Formulation for Graph Partitioning

2010

Math. Program.

103

If a mathematical program has many symmetric optima, solving it via Branchand-Bound techniques often yields search trees of disproportionate sizes; thus, finding and exploiting symmetries is an important task. We propose a method for automatically finding the formulation group of any given Mixed-Integer Nonlinear Program, and for reformulating the problem by means of static symmetry breaking constraints. The reformulated problem -which is likely to have fewer symmetric optima -can then be solved via standard Branch-and-Bound codes such as CPLEX (for linear programs) and COUENNE (for nonlinear programs). Our computational results include formulation group tables for the MIPLib3, MIPLib2003, GlobalLib and MINLPLib instance libraries and solution tables for some instances in the aforementioned libraries.

Section: Literature Reviewmentioning

confidence: 99%

Reformulations in mathematical programming: automatic symmetry detection and exploitation

2010

Math. Program.

103

“…[40]), appropriate global cuts (e.g. [60]) or special formulations [30,9] based on the problem structure. The main limitation of the methods in this category is that they are difficult to generalize and/or to be made automatic.…”

Section: Symmetry In Linearmentioning

confidence: 99%

“…(1, 2)(3, 4)(5, 6)(7, 8)(9, 10)(11, 12), (1, 3)(2, 4), (3, 5)(4, 6), (5, 7)(6, 8), (7,9) (8,10), (9,11) …”

Section: For Any H ≤ G If G(x)mentioning

confidence: 99%

Symmetry in Mathematical Programming

Mixed Integer Nonlinear Programming

2011

Abstract. Symmetry is mainly exploited in mathematical programming in order to reduce the computation times of enumerative algorithms. The most widespread approach rests on: (a) finding symmetries in the problem instance; (b) reformulating the problem so that it does not allow some of the symmetric optima; (c) solving the modified problem. Sometimes (b) and (c) are performed concurrently: the solution algorithm generates a sequence of subproblems, some of which are recognized to be symmetrically equivalent and either discarded or treated differently. We review symmetry-based analyses and methods for Linear Programming, Integer Linear Programming, Mixed-Integer Linear Programming and Semidefinite Programming. We then discuss a method (introduced in [35]) for automatically detecting symmetries of general (nonconvex) Nonlinear and Mixed-Integer Nonlinear Programming problems and a reformulation based on adjoining symmetry breaking constraints to the original formulation. We finally present a new theoretical and computational study of the formulation symmetries of the Kissing Number Problem.

“…Many different formulations of varying sizes where the clusters have maximal given cardinality have been proposed and compared in (Boulle 2004). A problem variant with node capacities has been studied in (Ferreira et al 1996).…”

Section: Graph Partitioningmentioning

confidence: 99%

Compact linearization for binary quadratic problems

2006

4OR

We show that a well-known linearization technique initially proposed for quadratic assignment problems can be generalized to a broader class of quadratic 0-1 mixed-integer problems subject to assignment constraints. The resulting linearized formulation is more compact and tighter than that obtained with a more usual linearization technique. We discuss the application of the compact linearization to three classes of problems in the literature, among which the graph partitioning problem.