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12
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
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 9 (7 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.
MixedInteger Nonlinear Optimization in Process Synthesis
, 1998
"... The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the process synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw ma ..."
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Cited by 7 (0 self)
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The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the process synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw materials into desired products. In recent years, the optimization approach to process synthesis has shown promise in tackling this challenge. It requires the development of a network of interconnected units, the process superstructure, that represents the alternative process flowsheets. The mathematical modeling of the superstructure has a mixed set of binary and continuous variables and results in a mixedinteger optimization model. Due to the nonlinearity of chemical models, these problems are generally classified as MixedInteger Nonlinear Programming (MINLP) problems. A number of local optimization algorithms, developed for the solution of this class of problems, are presented in this pap...
Nonconvex mixedinteger nonlinear programming: A survey
 Surveys in Operations Research and Management Science
, 2012
"... A wide range of problems arising in practical applications can be formulated as MixedInteger Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When nonconvexities are present, how ..."
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Cited by 7 (0 self)
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A wide range of problems arising in practical applications can be formulated as MixedInteger Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When nonconvexities are present, however, things become much more difficult, since then even the continuous relaxation is a global optimisation problem. We survey the literature on nonconvex MINLP, discussing applications, algorithms and software. Special attention is paid to the case in which the objective and constraint functions are quadratic. Key Words: mixedinteger nonlinear programming, global optimisation, quadratic programming, polynomial optimisation.
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 6 (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.
Recent Advances in Global Optimization for Process Synthesis, Design and Control: Enclosure of All Solutions
 Computers and Chemical Engineering
, 1999
"... Recent advances in global optimization for process synthesis, design and control are discussed. After a review of the chemical engineering contributions, we focus on the enclosure of all solutions of nonlinear constrained systems of equations. Important theoretical results are presented accompanied ..."
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Cited by 5 (0 self)
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Recent advances in global optimization for process synthesis, design and control are discussed. After a review of the chemical engineering contributions, we focus on the enclosure of all solutions of nonlinear constrained systems of equations. Important theoretical results are presented accompanied with computational studies on the enclosure of multiple steady states and all homogeneous azeotropes. 1 Introduction and Review A significant effort has been expended in the last four decades toward theoretical and algorithmic studies of applications that arise in Chemical Engineering Process Design, Process Synthesis, Process Control, as well as in Computational Chemistry and Molecular Biology. In the last decade we have experienced a dramatic growth of interest in Chemical Engineering for new methods of global optimization and their application to important engineering, as well as computational chemistry and molecular biology problems. Contributions from the chemical engineering communit...
Algorithmic Framework for Improving Heuristics in Stochastic, StageWise Optimization Problems
, 2004
"... ..."
Alain BILLIONNET (Examinateur), Professeur au Conservatoir National d’Arts et Métiers
, 2007
"... L’objet centrale de cette thèse est l’utilisation de techniques de reformulations en programmation mathématique. Les problèmes d’optimisation et de décision peuvent être décrits précisement par une formulation composée de: paramètres numériques, variables de decision (leur valeurs étant determinée ..."
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L’objet centrale de cette thèse est l’utilisation de techniques de reformulations en programmation mathématique. Les problèmes d’optimisation et de décision peuvent être décrits précisement par une formulation composée de: paramètres numériques, variables de decision (leur valeurs étant determinées grâce au resultat d’un procès algorithmique), une ou plusieurs fonctions objectives à optimiser, et plusieurs ensembles de contraintes; fonctions objectives et contraintes peuvent être exprimées explicitement comme functions de paramètres et variables, ou implicitement comme conditions sur les variables. Ces éléments, c’est à dire paramètres, variables, fonctions objectives et contraintes, forment un langage appélé programmation mathématique. Pour chaque problème donné d’optimisation ou décision, il y a d’habitude un nombre infini de différents formulations de programmation mathématique possibles. Selon l’algorithme utilisé pour les résoudre, formulations distinctes sont plus ou moins efficaces et/ou efficientes. En outre, plusieurs sousproblèmes ressortants de l’algorithme de solution peuvent euxmêmes être formulés comme des problèmes de programmation mathématique (appélés problèmes auxiliaires). Cette thèse présente un étude approfondi des transformations symboliques qui mappent des formulations de programmation mathématique a leur formes équivalentes et autres