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23
Reformulations in Mathematical Programming: A Computational Approach
"... Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathema ..."
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Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of blackbox functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization. 1
Reformulations in mathematical programming: Automatic symmetry detection and exploitation
 Mathematical Programming
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Formulation symmetries in circle packing
"... The performance of BranchandBound algorithms is severely impaired by the presence of symmetric optima in a given problem. We describe a method for the automatic detection of formulation symmetries in MINLP instances. A software implementation of this method is used to conjecture the group structur ..."
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The performance of BranchandBound algorithms is severely impaired by the presence of symmetric optima in a given problem. We describe a method for the automatic detection of formulation symmetries in MINLP instances. A software implementation of this method is used to conjecture the group structure of the problem symmetries of packing equal circles in a square. We provide a proof of the conjecture and compare the performance of spatial BranchandBound on the original problem with the performance on a reformulation that cuts away symmetric optima. Keywords: MINLP, spatial BranchandBound, Global Optimization, group, reformulation.
Feasibilitybased bounds tightening via fixed points
"... Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, whi ..."
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Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, which is known to be practically fast, and is thus deployed at every node of the search tree. From time to time, however, this technique fails to converge to its limit point in finite time, thereby slowing the whole BranchandBound search considerably. In this paper we propose a polynomial time method, based on solving a linear program, for computing the limit point of the Feasibility Based Bounds Tightening algorithm applied to linear equality and inequality constraints. Keywords: global optimization, MINLP, spatial BranchandBound, range reduction, constraint programming. 1
Symmetry in mathematical programming
 MIXED INTEGER NONLINEAR PROGRAMMING. VOLUME IMA
"... 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; ( ..."
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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 symmetrybased analyses and methods for Linear Programming, Integer Linear Programming, MixedInteger Linear Programming and Semidefinite Programming. We then discuss a method (introduced in [35]) for automatically detecting symmetries of general (nonconvex) Nonlinear and MixedInteger 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.
EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONS
"... Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We surv ..."
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Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of its most important applications, including molecular conformation, localization of sensor networks and statics. Key words. Matrix completion, barandjoint framework, graph rigidity, inverse problem, protein conformation, sensor network.
Reformulations in mathematical programming: Symmetry
 Mathematical Programming
, 2009
"... If a mathematical program (be it linear or nonlinear) has many symmetric optima, solving it via BranchandBound techniques often yields search trees of disproportionate sizes; thus, finding and exploiting symmetries is an important task. We propose a method for: (a) automatically finding the formul ..."
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If a mathematical program (be it linear or nonlinear) has many symmetric optima, solving it via BranchandBound techniques often yields search trees of disproportionate sizes; thus, finding and exploiting symmetries is an important task. We propose a method for: (a) automatically finding the formulation group of any given MixedInteger Nonlinear Program, and (b) reformulating the problem so that some symmetric solutions become infeasible. The reformulated problem can then be solved via standard BranchandBound 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, solution tables for some instances in the aforementioned libraries, and a theoretical and computational study of the symmetries of the Kissing Number Problem. 1
Automatic generation of symmetrybreaking constraints
"... Solution symmetries in integer linear programs often yield long BranchandBound based solution processes. We propose a method for finding elements of the permutation group of solution symmetries, and two different types of symmetrybreaking constraints to eliminate these symmetries at the modellin ..."
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Solution symmetries in integer linear programs often yield long BranchandBound based solution processes. We propose a method for finding elements of the permutation group of solution symmetries, and two different types of symmetrybreaking constraints to eliminate these symmetries at the modelling level. We discuss some preliminary computational results.