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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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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.
ReformulationLinearization Methods for Global Optimization
, 2007
"... Keywords: ReformulationLinearization Technique, liftandproject, tight relaxations, valid inequalities, model reformulation, convex hull, convex envelopes, mixedinteger 01 program, polynomial programs, nonconvex programs, factorable programs, reduced relaxations. Discrete and continuous nonconve ..."
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Cited by 4 (1 self)
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Keywords: ReformulationLinearization Technique, liftandproject, tight relaxations, valid inequalities, model reformulation, convex hull, convex envelopes, mixedinteger 01 program, polynomial programs, nonconvex programs, factorable programs, reduced relaxations. Discrete and continuous nonconvex programming problems arise in a host of practical applications in the context of production planning and control, locationallocation, distribution, economics and game theory, quantum chemistry, and process and engineering design situations. Several recent advances have been made in the development of branchandcut type algorithms for mixedinteger linear and nonlinear programming problems, as well as polyhedral outerapproximation methods for continuous nonconvex programming problems. At the heart of these approaches is a sequence of linear (or convex) programming relaxations that drive the solution process, and the success of such algorithms is strongly tied in with the strength or tightness of these relaxations. The ReformulationLinearizationTechnique (RLT) is a method that generates such tight linear programming relaxations for not only constructing exact solution algorithms, but also to design powerful heuristic procedures for large classes of discrete combinatorial and continuous nonconvex programming problems. Its development originated in [4, 5, 6], initially focusing on 01 and mixed 01 linear and
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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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.
WeC02.5 Mixed State Estimation for a Linear Gaussian Markov Model
"... Abstract — 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 ..."
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Abstract — 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. I.
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.