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15
Interval Analysis on Directed Acyclic Graphs for Global Optimization
 J. Global Optimization
, 2004
"... A directed acyclic graph (DAG) representation of optimization problems represents each variable, each operation, and each constraint in the problem formulation by a node of the DAG, with edges representing the ow of the computation. ..."
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Cited by 40 (8 self)
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A directed acyclic graph (DAG) representation of optimization problems represents each variable, each operation, and each constraint in the problem formulation by a node of the DAG, with edges representing the ow of the computation.
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.
Experiments with repeating weighted boosting search for optimization in signal processing applications
 IEEE Trans. Syst. Man Cybern. B, Cybern
, 2005
"... Abstract—Many signal processing applications pose optimization problems with multimodal and nonsmooth cost functions. Gradient methods are ineffective in these situations, and optimization methods that require no gradient and can achieve a global optimal solution are highly desired to tackle these d ..."
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Cited by 9 (6 self)
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Abstract—Many signal processing applications pose optimization problems with multimodal and nonsmooth cost functions. Gradient methods are ineffective in these situations, and optimization methods that require no gradient and can achieve a global optimal solution are highly desired to tackle these difficult problems. The paper proposes a guided global search optimization technique, referred to as the repeated weighted boosting search. The proposed optimization algorithm is extremely simple and easy to implement, involving a minimum programming effort. Heuristic explanation is given for the global search capability of this technique. Comparison is made with the two better known and widely used guided global search techniques, known as the genetic algorithm and adaptive simulated annealing, in terms of the requirements for algorithmic parameter tuning. The effectiveness of the proposed algorithm as a global optimizer are investigated through several application examples. Index Terms—Adaptive simulated annealing, boosting, evolutionary computation, genetic algorithm, global search, multistart, optimization, stochastic algorithm. I.
Validated linear relaxations and preprocessing: Some experiments, 2003. accepted for publication in
 SIAM J. Optim
"... Abstract. Based on work originating in the early 1970s, a number of recent global optimization algorithms have relied on replacing an original nonconvex nonlinear program by convex or linear relaxations. Such linear relaxations can be generated automatically through an automatic differentiation proc ..."
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Cited by 5 (3 self)
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Abstract. Based on work originating in the early 1970s, a number of recent global optimization algorithms have relied on replacing an original nonconvex nonlinear program by convex or linear relaxations. Such linear relaxations can be generated automatically through an automatic differentiation process. This process decomposes the objective and constraints (if any) into convex and nonconvex unary and binary operations. The convex operations can be approximated arbitrarily well by appending additional constraints, while the domain must somehow be subdivided (in an overall branchandbound process or in some other local process) to handle nonconvex constraints. In general, a problem can be hard if even a single nonconvex term appears. However, certain nonconvex terms lead to easiertosolve problems than others. Recently, Neumaier, Lebbah, Michel, ourselves, and others have paved the way to utilizing such techniques in a validated context. In this paper, we present a symbolic preprocessing step that provides a measure of the intrinsic difficulty of a problem. Based on this step, one of two methods can be chosen to relax nonconvex terms. This preprocessing step is similar to a method previously proposed by Epperly and Pistikopoulos [J. Global Optim., 11 (1997), pp. 287–311] for determining subspaces in which to branch, but we present it from a different point of view that is amenable to simplification of the problem presented to the linear programming solver, and within a validated context. Besides an illustrative example, we have implemented general relaxations in a validated context, as well as the preprocessing technique, and we present experiments on a standard test set. Finally, we present conclusions.
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...
Orthogonal packing of rectangles within isotropic convex regions
, 2009
"... A mixed integer continuous nonlinear model and a solution method for the problem of orthogonally packing identical rectangles within an arbitrary convex region are introduced in the present work. The convex region is assumed to be made of an isotropic material in such a way that arbitrary rotations ..."
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Cited by 2 (0 self)
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A mixed integer continuous nonlinear model and a solution method for the problem of orthogonally packing identical rectangles within an arbitrary convex region are introduced in the present work. The convex region is assumed to be made of an isotropic material in such a way that arbitrary rotations of the items, preserving the orthogonality constraint, are allowed. The solution method is based on a combination of branch and bound and activeset strategies for boundconstrained minimization of smooth functions. Numerical results show the reliability of the presented approach.
Predicting Molecular Structures: An Application of the Cutting Angle Method
 Phys. Chem. Chem. Phys
, 2003
"... this paper, we consider several wellknown molecular conformation problems to which we apply a new method of deterministic global optimization called the cutting angle method. We demonstrate that this method is competitive with other global optimization techniques for these molecular conformation ..."
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Cited by 1 (1 self)
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this paper, we consider several wellknown molecular conformation problems to which we apply a new method of deterministic global optimization called the cutting angle method. We demonstrate that this method is competitive with other global optimization techniques for these molecular conformation problems
Validated Constraint Solving – Practicalities, Pitfalls, and New Developments
"... Abstract. Many constraint propagation techniques iterate through the constraints in a straightforward manner, but can fail because they do not take account of the coupling between the constraints. However, some methods of taking account of this coupling are local in nature, and fail if the initial s ..."
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Cited by 1 (0 self)
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Abstract. Many constraint propagation techniques iterate through the constraints in a straightforward manner, but can fail because they do not take account of the coupling between the constraints. However, some methods of taking account of this coupling are local in nature, and fail if the initial search region is too large. We put into perspective newer methods, based on linear relaxations, that can often replace bruteforce search by solution of a large, sparse linear program. Robustness has been recognized as important in geometric computations and elsewhere for at least a decade, and more and more developers are including validation in the design of their systems. We provide citations to our work todate in developing validated versions of linear relaxations. This work is in the form of a brief review and prospectus for future development. We give various simple examples to illustrate our points. Keywords: constraint propagation, global optimization, linear relaxations, GlobSol
Deterministic Global Optimization of Molecular Structures Using Interval Analysis
, 2004
"... The search for the global minimum of a molecular potential energy surface is a challenging problem. The molecular structure corresponding to the global minimum is of particular importance since it usually dictates both the physical and chemical properties of the molecule. The existence of an extreme ..."
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Cited by 1 (1 self)
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The search for the global minimum of a molecular potential energy surface is a challenging problem. The molecular structure corresponding to the global minimum is of particular importance since it usually dictates both the physical and chemical properties of the molecule. The existence of an extremely large number of local minima, the number of which may increase exponentially with the size of the molecule, makes this global minimization problem extremely difficult. A new strategy is described here for solving such global minimization problems deterministically. The methodology is based on interval analysis, and provides a mathematical and computational guarantee that the molecular structure with the global minimum potential energy will be found. The technique is demonstrated using two sets of example problems. The first set involves a relatively simple potential model, and problems with up to 40 atoms. The second set involves a more realistic potential energy function, representative of those in current use, and problems with up to eleven atoms.
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.