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56
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 94 (22 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.
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 24 (19 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 24 (23 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.
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 23 (17 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.
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 23 (11 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.
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 16 (14 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.
G.: A good recipe for solving MINLPs
 Hybridizing metaheuristics and mathematical programming. Volume 10 of Annals of Information Systems
, 2009
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MixedInteger Nonlinear Programming Models and Algorithms for LargeScale Supply
 Chain Design with Stochastic Inventory Management. Industrial & Engineering Chemistry Research 2008
"... An important challenge for most chemical companies is to simultaneously consider inventory optimization and supply chain network design under demand uncertainty. This leads to a problem that requires integrating a stochastic inventory model with the supply chain network design model. This problem ca ..."
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Cited by 13 (7 self)
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An important challenge for most chemical companies is to simultaneously consider inventory optimization and supply chain network design under demand uncertainty. This leads to a problem that requires integrating a stochastic inventory model with the supply chain network design model. This problem can be formulated as a large scale combinatorial optimization model that includes nonlinear terms. Since these models are very difficult to solve, they require exploiting their properties and developing special solution techniques to reduce the computational effort. In this work, we analyze the properties of the basic model and develop solution techniques for a joint supply chain network design and inventory management model for a given product. The model is formulated as a nonlinear integer programming problem. By reformulating it as a mixedinteger nonlinear programming (MINLP) problem and using an associated convex relaxation model for initialization, we first propose a heuristic method to quickly obtain good quality solutions. Further, a decomposition algorithm based on Lagrangean relaxation is developed for obtaining global or nearglobal optimal solutions. Extensive computational examples with up to 150 distribution centers and 150 retailers are presented to illustrate the performance of the algorithms and to compare them with the fullspace solution. To whom all correspondence should be addressed.
On convex relaxations of quadrilinear terms
, 2009
"... The best known method to find exact or at least εapproximate solutions to polynomial programming problems is the spatial BranchandBound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are o ..."
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Cited by 11 (8 self)
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The best known method to find exact or at least εapproximate solutions to polynomial programming problems is the spatial BranchandBound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the original program. Although convex envelopes are explicitly known (via linear inequalities) for bilinear and trilinear terms on arbitrary boxes, such a description is unknown, in general, for multilinear terms of higher order. In this paper, we study convex relaxations of quadrilinear terms. We exploit associativity to rewrite such terms as products of bilinear and trilinear terms. Using a general technique, we establish that, any relaxation for klinear terms that employs a successive use of relaxing bilinear terms (via the bilinear convex envelope) can be improved by employing instead a relaxation of a trilinear term (via the trilinear convex envelope). We present a computational analysis which helps establish which relaxations are strictly tighter, and we apply our findings to two wellstudied applications: the Molecular Distance Geometry Problem and the HartreeFock Problem.