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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 ..."
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Cited by 59 (9 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 BoxConstrained Optimization Algorithm With Negative Curvature Directions and Spectral Projected Gradients
, 2001
"... A practical algorithm for boxconstrained optimization is introduced. The algorithm combines an activeset strategy with spectral projected gradient iterations. In the interior of each face a strategy that deals eciently with negative curvature is employed. Global convergence results are given. ..."
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Cited by 28 (5 self)
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A practical algorithm for boxconstrained optimization is introduced. The algorithm combines an activeset strategy with spectral projected gradient iterations. In the interior of each face a strategy that deals eciently with negative curvature is employed. Global convergence results are given. Numerical results are presented. Keywords: box constrained minimization, active set methods, spectral projected gradients, dogleg path methods. AMS Subject Classication: 49M07, 49M10, 65K, 90C06, 90C20. 1
On the Boundedness of Penalty Parameters in an Augmented Lagrangian Method with Constrained Subproblems
, 2011
"... Augmented Lagrangian methods are effective tools for solving largescale nonlinear programming problems. At each outer iteration a minimization subproblem with simple constraints, whose objective function depends on updated Lagrange multipliers and penalty parameters, is approximately solved. When t ..."
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Cited by 6 (1 self)
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Augmented Lagrangian methods are effective tools for solving largescale nonlinear programming problems. At each outer iteration a minimization subproblem with simple constraints, whose objective function depends on updated Lagrange multipliers and penalty parameters, is approximately solved. When the penalty parameter becomes very large the subproblem is difficult, therefore the effectiveness of this approach is associated with boundedness of penalty parameters. In this paper it is proved that, under more natural assumptions than the ones up to now employed, penalty parameters are bounded. For proving the new boundedness result, the original algorithm has been slightly modified. Numerical consequences of the modifications are discussed and computational experiments are presented.
New formulations for the kissing number problem
 Discrete Applied Mathematics
"... Determining the maximum number of Ddimensional spheres of radius r that can be adjacent to a central sphere of radius r is known as the Kissing Number Problem (KNP). The problem has been solved for 2, 3 and very recently for 4 dimensions. We present two nonlinear (nonconvex) mathematical programmin ..."
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Cited by 5 (4 self)
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Determining the maximum number of Ddimensional spheres of radius r that can be adjacent to a central sphere of radius r is known as the Kissing Number Problem (KNP). The problem has been solved for 2, 3 and very recently for 4 dimensions. We present two nonlinear (nonconvex) mathematical programming models for the solution of the KNP. We solve the problem by using two stochastic global optimization methods: a Multi Level Single Linkage algorithm and a Variable Neighbourhood Search. We obtain numerical results for 2, 3 and 4 dimensions.
Nonlinearprogramming reformulation of the Ordervalue optimization problem
 Mathematical Methods of Operations Research 61
, 2005
"... Ordervalue optimization (OVO) is a generalization of the minimax problem motivated by decisionmaking problems under uncertainty and by robust estimation. New optimality conditions for this nonsmooth optimization problem are derived. An equivalent mathematical programming problem with equilibrium c ..."
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Cited by 4 (2 self)
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Ordervalue optimization (OVO) is a generalization of the minimax problem motivated by decisionmaking problems under uncertainty and by robust estimation. New optimality conditions for this nonsmooth optimization problem are derived. An equivalent mathematical programming problem with equilibrium constraints is deduced. The relation between OVO and this nonlinearprogramming reformulation is studied. Particular attention is given to the relation between local minimizers and stationary points of both problems.
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 4 (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:
Minimization of Discontinuous Cost Functions by Smoothing
, 2001
"... The minimization of functions that include discontinuous costs is considered. Sufficient conditions are introduced under which a smoothing scheme converges to the solution of the original problem. Some examples are presented and the numerical behavior of the proposed approach is commented. ..."
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The minimization of functions that include discontinuous costs is considered. Sufficient conditions are introduced under which a smoothing scheme converges to the solution of the original problem. Some examples are presented and the numerical behavior of the proposed approach is commented.
Boxconstrained minimization reformulations of complementarity problems in secondorder cones ∗
, 2006
"... Reformulations of a generalization of a secondorder cone complementarity problem (GSOCCP) as optimization problems are introduced, that preserve differentiability. Equivalence results are proved in the sense that the global minimizers of the reformulations with zero objective value are solutions to ..."
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Reformulations of a generalization of a secondorder cone complementarity problem (GSOCCP) as optimization problems are introduced, that preserve differentiability. Equivalence results are proved in the sense that the global minimizers of the reformulations with zero objective value are solutions to the GSOCCP and vice versa. Since the optimization problems involved include only simple constraints, a whole range of minimization algorithms may be used to solve the equivalent problems. Taking into account that optimization algorithms usually seek stationary points, a theoretical result is established that ensures equivalence between stationary points of the reformulation and solutions to the GSOCCP. Numerical experiments are presented that illustrate the advantages and disadvantages of the reformulations.