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13
Global minimization using an Augmented Lagrangian method with variable lowerlevel constraints
, 2007
"... A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εkglobal minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global c ..."
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Cited by 21 (1 self)
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A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εkglobal minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an εglobal minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.
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.
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.
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
Efficient Pruning Technique Based on Linear Relaxations
, 2005
"... This paper extends the Quadfiltering algorithm for handling general nonlinear systems. This extended algorithm is based on the RLT (ReformulationLinearization Technique) schema. In the reformulation phase, tight convex and concave approximations of nonlinear terms are generated, that's to say for ..."
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Cited by 4 (0 self)
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This paper extends the Quadfiltering algorithm for handling general nonlinear systems. This extended algorithm is based on the RLT (ReformulationLinearization Technique) schema. In the reformulation phase, tight convex and concave approximations of nonlinear terms are generated, that's to say for bilinear terms, product of variables, power and univariate terms. New variables are introduced to linearize the initial constraint system. A linear programming solver is called to prune the domains. A combination of this filtering technique with Boxconsistency filtering algorithm has been investigated. Experimental results on difficult problems show that a solver based on this combination outperforms classical CSP solvers.
Optimization of Multiuser MIMO Networks with Interference
"... Abstract — Maximizing the total mutual information of a multiuser multipleinput multipleoutput (MIMO) system with interference is a wellknown and challenging problem. In this paper, we consider the power control problem of finding the maximum sum of mutual information for multiuser MIMO systems w ..."
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Cited by 2 (0 self)
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Abstract — Maximizing the total mutual information of a multiuser multipleinput multipleoutput (MIMO) system with interference is a wellknown and challenging problem. In this paper, we consider the power control problem of finding the maximum sum of mutual information for multiuser MIMO systems with equal power allocation at each link. A new and powerful global optimization method using a branchandbound framework coupled with the reformulationlinearization technique (BB/RLT) is introduced. The proposed BB/RLT is the first such method that guarantees finding a global optimum for multiuser MIMO systems with interference. In addition, we propose a modified branchandbound (BB) variable selection strategy to accelerate the convergence process, and apply the proposed technique to several MIMO systems in order to demonstrate its efficacy. I.
Global Nonlinear Programming with possible infeasibility and finite termination
, 2012
"... In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the αBB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In th ..."
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Cited by 1 (0 self)
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In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the αBB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In the present research, the algorithm mentioned above will be improved in several crucial aspects. On the one hand, feasibility of the problem will not be required. Possible infeasibility will be detected in finite time by the new algorithms and optimal infeasibility results will be proved. On the other hand, finite termination results thatguaranteeoptimalityand/orfeasibilityuptoanyrequiredprecisionwillbeprovided. An adaptive modification in which subproblem tolerances depend on current feasibility and complementarity will also be given. The adaptive algorithm allows the augmented Lagrangian subproblems to be solved without requiring unnecessary potentially high precisions in the intermediate steps of the method, which improves the overall efficiency. Experiments showing how the new algorithms and results are related to practical computations will be given.
Université d’Oran EsSenia, Faculté des Sciences,
"... Abstract. This paper extends the Quadfiltering algorithm for handling general nonlinear systems. This extended algorithm is based on the RLT (ReformulationLinearization Technique) schema. In the reformulation phase, tight convex and concave approximations of nonlinear terms are generated, that’s t ..."
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Abstract. This paper extends the Quadfiltering algorithm for handling general nonlinear systems. This extended algorithm is based on the RLT (ReformulationLinearization Technique) schema. In the reformulation phase, tight convex and concave approximations of nonlinear terms are generated, that’s to say for bilinear terms, product of variables, power and univariate terms. New variables are introduced to linearize the initial constraint system. A linear programming solver is called to prune the domains. A combination of this filtering technique with Boxconsistency filtering algorithm has been investigated. Experimental results on difficult problems show that a solver based on this combination outperforms classical CSP solvers. 1