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27
LargeScale ActiveSet BoxConstrained Optimization Method with Spectral Projected Gradients
 Computational Optimization and Applications
, 2001
"... A new activeset method for smooth boxconstrained minimization is introduced. The algorithm combines an unconstrained method, including a new linesearch which aims to add many constraints to the working set at a single iteration, with a recently introduced technique (spectral projected gradien ..."
Abstract

Cited by 60 (11 self)
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A new activeset method for smooth boxconstrained minimization is introduced. The algorithm combines an unconstrained method, including a new linesearch which aims to add many constraints to the working set at a single iteration, with a recently introduced technique (spectral projected gradient) for dropping constraints from the working set. Global convergence is proved. A computer implementation is fully described and a numerical comparison assesses the reliability of the new algorithm. Keywords: Boxconstrained minimization, numerical methods, activeset strategies, Spectral Projected Gradient. 1
A new active set algorithm for box constrained optimization
 SIAM J. Optim
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Optimizing the Packing of Cylinders into a Rectangular Container: A Nonlinear Approach
, 2003
"... The container loading problem has important industrial and commercial applications. An increase in the number of items in a container leads to a decrease in cost. For this reason the related optimization problem is of economic importance. In this work, a procedure based on a nonlinear decision pr ..."
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Cited by 22 (2 self)
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The container loading problem has important industrial and commercial applications. An increase in the number of items in a container leads to a decrease in cost. For this reason the related optimization problem is of economic importance. In this work, a procedure based on a nonlinear decision problem to solve the cylinder packing problem with identical diameters is presented. This formulation is based on the fact that the centers of the cylinders have to be inside the rectangular box de ned by the base of the container (a radius far from the frontier) and far from each other at least one diameter. With this basic premise the procedure tries to nd the maximum number of cylinder centers that satisfy these restrictions. The continuous nature of the problem is one of the reasons that motivated this study. A comparative study with other methods of the literature is presented and better results are achieved.
Minimizing the object dimensions in circle and sphere packing problems
, 2006
"... Given a fixed set of identical or differentsized circular items, the problem we deal withconsists on finding the smallest object within which the items can be packed. Circular, triangular, squared, rectangular and also strip objects are considered. Moreover, 2D and3D problems are treated. Twiced ..."
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Cited by 16 (1 self)
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Given a fixed set of identical or differentsized circular items, the problem we deal withconsists on finding the smallest object within which the items can be packed. Circular, triangular, squared, rectangular and also strip objects are considered. Moreover, 2D and3D problems are treated. Twicedifferentiable models for all these problems are presented. A strategy to reduce the complexity of evaluating the models is employed and, as a consequence, instances with a large number of items can be considered. Numerical experiments show the flexibility and reliability of the new unified approach.
Improving ultimate convergence of an Augmented Lagrangian method
, 2007
"... Optimization methods that employ the classical PowellHestenesRockafellar Augmented Lagrangian are useful tools for solving Nonlinear Programming problems. Their reputation decreased in the last ten years due to the comparative success of InteriorPoint Newtonian algorithms, which are asymptoticall ..."
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Cited by 13 (0 self)
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Optimization methods that employ the classical PowellHestenesRockafellar Augmented Lagrangian are useful tools for solving Nonlinear Programming problems. Their reputation decreased in the last ten years due to the comparative success of InteriorPoint Newtonian algorithms, which are asymptotically faster. In the present research a combination of both approaches is evaluated. The idea is to produce a competitive method, being more robust and efficient than its “pure” counterparts for critical problems. Moreover, an additional hybrid algorithm is defined, in which the Interior Point method is replaced by the Newtonian resolution of a KKT system identified by the Augmented Lagrangian algorithm. The software used in this work is freely available through the Tango Project web page:
Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization
, 2005
"... The orthogonal packing of rectangular items in an arbitrary convex region is consideredin this work. The packing problem is modeled as the problem of deciding for the feasibility or infeasibility of a set of nonlinear equality and inequality constraints. A procedure basedon nonlinear programming i ..."
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Cited by 11 (2 self)
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The orthogonal packing of rectangular items in an arbitrary convex region is consideredin this work. The packing problem is modeled as the problem of deciding for the feasibility or infeasibility of a set of nonlinear equality and inequality constraints. A procedure basedon nonlinear programming is introduced and numerical experiments which show that the new procedure is reliable are exhibited. Scope and purpose: We address the problem of packing orthogonal rectangles within anarbitrary convex region. We aim to show that smooth nonlinear programming models are a reliable alternative for packing problems and that wellknown generalpurpose methodsbased on continuous optimization can be used to solve the models. Numerical experiments illustrate the capabilities and limitations of the approach.
Method of Sentinels for Packing Items within Arbitrary Convex Regions
, 2005
"... A new method is introduced for packing items in convex regions of the Euclidian ndimensional space. By means of this approach the packing problem becomes a global finitedimensional continuous optimization problem. The strategy is based on the new concept of sentinels. Sentinels sets are finite subse ..."
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Cited by 8 (2 self)
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A new method is introduced for packing items in convex regions of the Euclidian ndimensional space. By means of this approach the packing problem becomes a global finitedimensional continuous optimization problem. The strategy is based on the new concept of sentinels. Sentinels sets are finite subsets of the items to be packed such that, when two items are superposed, at least one sentinel of one item is in the interior of the other. Minimal sets of sentinels are found in simple 2−dimensional cases. Numerical experiments and pictures showing the potentiality of the new technique are presented.
Partial Spectral Projected Gradient Method with ActiveSet Strategy for Linearly Constrained Optimization
, 2009
"... A method for linearly constrained optimization which modifies and generalizes recent boxconstraint optimization algorithms is introduced. The new algorithm is based on a relaxed form of Spectral Projected Gradient iterations. Intercalated with these projected steps, internal iterations restricted t ..."
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Cited by 7 (0 self)
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A method for linearly constrained optimization which modifies and generalizes recent boxconstraint optimization algorithms is introduced. The new algorithm is based on a relaxed form of Spectral Projected Gradient iterations. Intercalated with these projected steps, internal iterations restricted to faces of the polytope are performed, which enhance the efficiency of the algorithms. Convergence proofs are given and numerical experiments are included and commented. Software supporting this paper is available through the Tango
Secondorder negativecurvature methods for boxconstrained and general constrained optimization
, 2009
"... A Nonlinear Programming algorithm that converges to secondorder stationary points is introduced in this paper. The main tool is a secondorder negativecurvature method for boxconstrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is ..."
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Cited by 6 (0 self)
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A Nonlinear Programming algorithm that converges to secondorder stationary points is introduced in this paper. The main tool is a secondorder negativecurvature method for boxconstrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (PowellHestenesRockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to secondorder stationary points in situations in which firstorder methods fail are exhibited.