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Reformulations in Mathematical Programming: A Computational Approach
"... 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 ex ..."
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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.
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 21 (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.
ReformulationLinearization methods for global optimization. Available from: http://www.lix.polytechnique.fr/liberti/rlt_encopt2.pdf
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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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Cited by 6 (1 self)
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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.
Mixed State Estimation for a Linear Gaussian Markov Model
, 2008
"... We consider a discretetime dynamical system with Boolean and continuous states, with the continuous state propagating linearly in the continuous and Boolean state variables, and an additive Gaussian process noise, and where each Boolean state component follows a simple Markov chain. This model, wh ..."
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Cited by 4 (4 self)
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We consider a discretetime dynamical system with Boolean and continuous states, with the continuous state propagating linearly in the continuous and Boolean state variables, and an additive Gaussian process noise, and where each Boolean state component follows a simple Markov chain. This model, which can be considered a hybrid or jumplinear system with very special form, or a standard linear GaussMarkov dynamical system driven by a Boolean Markov process, arises in dynamic fault detection, in which each Boolean state component represents a fault that can occur. We address the problem of estimating the state, given Gaussian noise corrupted linear measurements. Computing the exact maximum a posteriori (MAP) estimate entails solving a mixed integer quadratic program, which is computationally difficult in general, so we propose an approximate MAP scheme, based on a convex relaxation, followed by rounding and (possibly) further local optimization. Our method has a complexity that grows linearly in the time horizon and cubicly with the state dimension, the same as a standard Kalman filter. Numerical experiments suggest that it performs very well in practice.
Techniques de reformulation en . . .
, 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
Solving a Quantum Chemistry problem with deterministic Global Optimization
, 2005
"... The HartreeFock method is well known in quantum chemistry, and widely used to obtain atomic and molecular eletronic wave functions, based on the minimization of a functional of the energy. This gives rise to a multiextremal, nonconvex, polynomial optimization problem. We give a novel mathematical ..."
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The HartreeFock method is well known in quantum chemistry, and widely used to obtain atomic and molecular eletronic wave functions, based on the minimization of a functional of the energy. This gives rise to a multiextremal, nonconvex, polynomial optimization problem. We give a novel mathematical programming formulation of the problem, which we solve by using a spatial BranchandBound algorithm. Lower bounds are obtained by solving a tight linear relaxation of the problem derived from an exact reformulation based on reduction constraints (a subset of RLT constraints). The proposed approach was successfully applied to the groundstate of the He and Be atoms.
Optimization problems arising in software architecture
, 2006
"... Software architecture is the process of planning and designing a largescale software, and a fundamental industrial discipline within the field of software engineering. The current body of knowledge in software architecture is a mixture of personal experience and precise methods. In this paper we m ..."
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Software architecture is the process of planning and designing a largescale software, and a fundamental industrial discipline within the field of software engineering. The current body of knowledge in software architecture is a mixture of personal experience and precise methods. In this paper we move a step towards the formalization of this discipline by describing some optimization problems that arise in the field. 1
NONLINEAR OPTIMAL CONTROL APPROACH TO SCHEDULING PROBLEMS
"... Abstract: Planning and scheduling are activities of major economic importance. Among various planning and scheduling problems, this paper focuses on the scheduling of crude oil and petroleum derivatives in ports: the jetty scheduling problem. The problem has combinatorial (allocation, sorting) and n ..."
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Abstract: Planning and scheduling are activities of major economic importance. Among various planning and scheduling problems, this paper focuses on the scheduling of crude oil and petroleum derivatives in ports: the jetty scheduling problem. The problem has combinatorial (allocation, sorting) and nonlinear (product blending) aspects, and multiobjective goals (minimization of different costs). We consider problems composed by tankers, tanks, pipelines, and jetties. Simulation models have been used as basis for this formulation. Moreover, reallife instances of the problem are of large scale and, if one wants to solve them in the daily operation of a harbor, they must be solved within minutes in common computer stations. This article proposes an original approach based on a continuous nonlinear optimal control model, without integer (binary) decision variables, which can reduce the problem’s size. Numerical experiments are presented using efficient NLP methods, such as LSGRG2, MINOS, and SNOPT.