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49
Review of nonlinear mixedinteger and disjunctive programming techniques
 Optimization and Engineering
, 2002
"... This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, OuterApproximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are ex ..."
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Cited by 61 (15 self)
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This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, OuterApproximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.
An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms
 Journal of Global Optimization
, 2005
"... Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subse ..."
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Cited by 19 (10 self)
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Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subset of ReformulationLinearization Technique constraints) which do not a#ect the feasible region of the original NLP but tighten that of its convex relaxation to the extent that some bilinear terms may be dropped from the problem formulation. We present an e#cient graphtheoretical algorithm for e#ecting such exact reformulations of large, sparse NLPs. The global solution of the reformulated problem using spatial Branchand Bound algorithms is usually significantly faster than that of the original NLP. We illustrate this point by applying our algorithm to a set of pooling and blending global optimization problems.
REFORMULATIONS IN MATHEMATICAL PROGRAMMING: DEFINITIONS AND SYSTEMATICS
, 2008
"... A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations c ..."
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Cited by 19 (14 self)
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A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations can be carried out automatically. Reformulation techniques are very common in mathematical programming but interestingly they have never been studied under a common framework. This paper attempts to move some steps in this direction. We define a framework for storing and manipulating mathematical programming formulations, give several fundamental definitions categorizing reformulations in essentially four types (optreformulations, narrowings, relaxations and approximations). We establish some theoretical results and give reformulation examples for each type.
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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Cited by 18 (13 self)
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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
Molecular distance geometry methods: From continuous to discrete
, 2009
"... Distance geometry problems arise from the need to position entities in the Euclidean Kspace given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph (graph drawing). In the context ..."
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Cited by 18 (18 self)
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Distance geometry problems arise from the need to position entities in the Euclidean Kspace given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph (graph drawing). In the context of molecular distance geometry, the distances are usually known because of chemical properties and Nuclear Magnetic Resonance experiments; sensor networks can estimate their relative distance by recording the power loss during a twoway exchange; finally, when drawing graphs in 2D or 3D, the graph to be drawn is given, and therefore distances between vertices can be computed. Distance geometry problems involve a search in a continuous Euclidean space, but sometimes the problem structure helps reduce the search to a discrete set of points. In this paper we survey some continuous and discrete methods for solving some problems of molecular distance geometry. 1
A good recipe for solving MINLPs
 Proceedings of Metaheuristics 2008 Conference, Bertinoro, 2008. Università di
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Computational Experience With The Molecular Distance Geometry Problem
"... In this work we consider the molecular distance geometry problem, which can be defined as the determination of the threedimensional structure of a molecule based on distances between some pairs of atoms. We address the problem as a nonconvex leastsquares problem. We apply three global optimization ..."
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Cited by 11 (10 self)
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In this work we consider the molecular distance geometry problem, which can be defined as the determination of the threedimensional structure of a molecule based on distances between some pairs of atoms. We address the problem as a nonconvex leastsquares problem. We apply three global optimization algorithms (spatial BranchandBound, Variable Neighbourhood Search, Multi Level Single Linkage) to two sets of instances, one taken from the literature and the other new. Keywords: molecular conformation, distance geometry, global optimization, spatial BranchandBound, variable neighbourhood search, multi level single linkage.
Reformulation and Convex Relaxation Techniques for Global Optimization
 4OR
, 2004
"... Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested i ..."
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Cited by 10 (8 self)
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Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested in determining the globally optimal point. This thesis is concerned with techniques for establishing such global optima using spatial BranchandBound (sBB) algorithms.
Comparison of Deterministic and Stochastic Approaches to global optimization
"... In this paper we compare two different approaches to nonconvex global optimization. The first one is a deterministic spatial BranchandBound algorithm (sBB), whereas the second approach is a quasi Monte Carlo (QMC) variant of a stochastic multi level single linkage (MLSL) algorithm. Both algorithms ..."
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Cited by 8 (3 self)
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In this paper we compare two different approaches to nonconvex global optimization. The first one is a deterministic spatial BranchandBound algorithm (sBB), whereas the second approach is a quasi Monte Carlo (QMC) variant of a stochastic multi level single linkage (MLSL) algorithm. Both algorithms apply to problems in a very general form and are not dependent on problem structure. The test suite we chose is fairly extensive in scope, in that it includes constrained and unconstrained problems, continuous and mixedinteger problems. The conclusion of the tests is that in general the QMC variant of the MLSL algorithm is more efficient, although in some instances the BranchandBound algorithm is capable of locating the global optimum of hard problems in just one iteration.