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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 55 (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.
Global Optimization of MixedInteger Nonlinear Programs: A Theoretical and Computational Study
 Mathematical Programming
, 2003
"... This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixedinteger nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototy ..."
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Cited by 51 (1 self)
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This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixedinteger nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branchandbound algorithm. In the theoretical...
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 19 (17 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
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 17 (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
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 17 (13 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.
Convex Extensions and Envelopes of Lower SemiContinuous Functions
, 2000
"... We define a convex extension of a lowersemicontinuous function to be a convex function that is identical to the given function over a prespecified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions under which a convex extension can be cons ..."
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Cited by 16 (1 self)
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We define a convex extension of a lowersemicontinuous function to be a convex function that is identical to the given function over a prespecified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex extension in a welldefined sense. Using the notion of a generating set, we establish conditions under which the tightest convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of x/y over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outerlinearization of bilinear terms.
A Finite Branch and Bound Algorithm for TwoStage Stochastic Integer Programs
, 2000
"... This paper addresses a general class of twostage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the c ..."
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Cited by 15 (4 self)
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This paper addresses a general class of twostage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Our computational results indicate superior performance of the proposed algorithm in comparison to the existing literature. Keywords: stochastic integer programming, branch and bound, finite algorithms. 1 Introduction Under the twostage stochastic programming paradigm, the decision variables of an optimization problem under uncertainty are partitioned into two sets. The first stage variables are those that have to be decided before the actual realization of the uncertain parameters. Subsequently, once the random events have presented themselves, further design or operational ...
A good recipe for solving MINLPs
"... Abstract. Finding good (or even just feasible) solutions for MixedInteger Nonlinear Programming problems independently of the specific problem structure is a very hard but practically useful task, specially when the objective/constraints are nonconvex. We present a generalpurpose heuristic based on ..."
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Cited by 14 (13 self)
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Abstract. Finding good (or even just feasible) solutions for MixedInteger Nonlinear Programming problems independently of the specific problem structure is a very hard but practically useful task, specially when the objective/constraints are nonconvex. We present a generalpurpose heuristic based on Variable Neighbourhood Search, Local Branching, Sequential Quadratic Programming and BranchandBound. We test the proposed approach on the MINLPLib, discussing optimality, reliability and speed. 1
A Global Optimization Algorithm for Nonconvex Generalized Disjunctive Programming and Applications to Process Systems
 Computers and Chemical Engineering
, 2000
"... A global optimization algorithm for nonconvex Generalized Disjunctive Programming (GDP) problems is proposed in this paper. By making use of convex underestimating functions for bilinear, linear fractional and concave separable functions in the continuous variables, the convex hull of each nonlinear ..."
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Cited by 14 (8 self)
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A global optimization algorithm for nonconvex Generalized Disjunctive Programming (GDP) problems is proposed in this paper. By making use of convex underestimating functions for bilinear, linear fractional and concave separable functions in the continuous variables, the convex hull of each nonlinear disjunction is constructed. The relaxed convex GDP problem is then solved in the first level of a twolevel branch and bound algorithm, in which a discrete branch and bound search is performed on the disjunctions to predict lower bounds. In the second level, a spatial branch and bound method is used to solve nonconvex NLP problems for updating the upper bound. The proposed algorithm exploits the convex hull relaxation for the discrete search, and the fact that the spatial branch and bound is restricted to fixed discrete variables in order to predict tight lower bounds. Application of the proposed algorithm to several example problems is shown, as well as comparisons with other algorithms.
Mathematical Programs with Equilibrium Constraints: Automatic Reformulation and Solution via Constrained Optimization
, 2002
"... Constrained optimization has been extensively used to... This paper briefly reviews some methods available to solve these problems and describes a new suite of tools for working with MPEC models. Computational results demonstrating... ..."
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Cited by 13 (3 self)
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Constrained optimization has been extensively used to... This paper briefly reviews some methods available to solve these problems and describes a new suite of tools for working with MPEC models. Computational results demonstrating...