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Nonmonotone spectral projected gradient methods on convex sets
 SIAM Journal on Optimization
, 2000
"... Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo–Lampariello–Lucidi nonmonotone lin ..."
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Cited by 135 (25 self)
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Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo–Lampariello–Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the onedimensional search process. Convergence properties and extensive numerical results are presented.
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 Spectral Conjugate Gradient Method for Unconstrained Optimization
, 1999
"... A family of scaled conjugategradient algorithms for largescale unconstrained minimization is dened. The Perry, the PolakRibiere and the FletcherReeves formulae are compared using a spectral scaling derived from Raydan's spectral gradient optimization method. The best combination of formula, ..."
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Cited by 13 (1 self)
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A family of scaled conjugategradient algorithms for largescale unconstrained minimization is dened. The Perry, the PolakRibiere and the FletcherReeves formulae are compared using a spectral scaling derived from Raydan's spectral gradient optimization method. The best combination of formula, scaling and initial choice of steplength is compared against well known algorithms using a classical set of problems. An additional comparison involving an illconditioned estimation problem in Optics is presented. Keywords. Unconstrained minimization, spectral gradient method, conjugate gradients. AMS: 49M07, 49M10, 90C06, 65K. 1
Augmented Lagrangian algorithms based on the spectral projected gradient method for solving nonlinear programming problems
"... The Spectral Projected Gradient method (SPG) is an algorithm for largescale boundconstrained optimization introduced recently by Birgin, Martnez and Raydan. It is based on Raydan's unconstrained generalization of the BarzilaiBorwein method for quadratics. The SPG algorithm turned out to be surpri ..."
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Cited by 12 (3 self)
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The Spectral Projected Gradient method (SPG) is an algorithm for largescale boundconstrained optimization introduced recently by Birgin, Martnez and Raydan. It is based on Raydan's unconstrained generalization of the BarzilaiBorwein method for quadratics. The SPG algorithm turned out to be surprisingly eective for solving many largescale minimization problems with box constraints. Therefore, it is natural to test its performance for solving the subproblems that appear in nonlinear programming methods based on augmented Lagrangians. In this work, augmented Lagrangian methods which use SPG as underlying convexconstraint solver are introduced (ALSPG), and the methods are tested in two sets of problems. First, a meaningful subset of largescale nonlinearly constrained problems of the CUTE collection is solved and compared with the performance of LANCELOT. Second, a family of location problems in the minimax formulation is solved against the package FFSQP.
Gradient method with dynamical retards for largescale optimization problems
 Electronic Transactions on Numerical Analysis (ETNA
, 2003
"... Abstract. We consider a generalization of the gradient method with retards for the solution of largescale unconstrained optimization problems. Recently, the gradient method with retards was introduced to find global minimizers of largescale quadratic functions. The most interesting feature of this ..."
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Cited by 3 (1 self)
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Abstract. We consider a generalization of the gradient method with retards for the solution of largescale unconstrained optimization problems. Recently, the gradient method with retards was introduced to find global minimizers of largescale quadratic functions. The most interesting feature of this method is that it does not involve a decrease in the objective function, which allows fast local convergence. On the other hand, nonmonotone globalization strategies, that preserve local behavior for the nonquadratic case, have proved to be very effective when associated with low storage methods. In this work, the gradient method with retards is generalized and combined in a dynamical way with nonmonotone globalization strategies to obtain a new method for minimizing nonquadratic functions, that can deal efficiently with large problems. Encouraging numerical experiments on wellknown test problems are presented. Key words. spectral gradient method, nonmonotone line search, BarzilaiBorwein method, PolakRibière method, Rayleigh quotient.
Minimization Subproblems and Heuristics for an Applied Clustering Problem
, 2001
"... A practical problem that requires the classification of a set of points of R^n using a criterion not sensitive to bounded outliers is studied in this paper. A fixedpoint (kmeans) algorithm is defined that uses an arbitrary distance function. Finite convergence is proved. A robust distance defined ..."
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Cited by 2 (1 self)
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A practical problem that requires the classification of a set of points of R^n using a criterion not sensitive to bounded outliers is studied in this paper. A fixedpoint (kmeans) algorithm is defined that uses an arbitrary distance function. Finite convergence is proved. A robust distance defined by Boente, Fraiman and Yohai is selected for applications. Smooth approximations of this distance are defined and suitable heuristics are introduced to enhance the probability of finding global optimizers. A reallife example is presented and commented.
A Fast and Global Two Point Low Storage Optimization Technique for Tracing Rays
"... We present the problem of tracing rays in 2D and 3D heterogeneous isotropic media as a set of optimization problems. Each optimization problem is obtained by applying Fermat's principle to an approximation of the travel time equation from a fixed source to a fixed receiver. We assume a piecewise lin ..."
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We present the problem of tracing rays in 2D and 3D heterogeneous isotropic media as a set of optimization problems. Each optimization problem is obtained by applying Fermat's principle to an approximation of the travel time equation from a fixed source to a fixed receiver. We assume a piecewise linear raypath which simplifies the computations of the problem, in the same way Mao and Stuart suggested in a very recent paper. Here, instead, the reflectors geometry and the velocity function are computed by using nonuniformly biharmonic splines. This biharmonic spline is easy to apply to interpolation problems in three and more dimensions and it also has minimum curvature. On the other hand, to solve the optimization problem we use the Global Spectral Gradient method. This recent developed optimization scheme is a low storage optimization technique that requires very few floating point operations. It only requires the gradient of the travel time function, and it does not require a close initial raypath. These three properties of the optimization method and the assumption of piecewise linear rays make this ray tracing scheme a very fast and effective method when estimating velocities via tomography. Besides, in a 2D or 3D stratified homogeneous medium, we show that this ray tracing approach converges to the global minimum, if the optimization problem has a solution. We present some numerical results that indicate that this scheme outperforms some classical optimization methods recently used for solving the same problem.
Spectral Gradient Methods for Linearly Constrained
"... Linearly constrained optimization problems with simple bounds are considered in the present work. First, a preconditioned spectral gradient method is de ned for the case in which no simple bounds are present. This algorithm can be viewed as a quasiNewton method in which the approximate Hessians ..."
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Linearly constrained optimization problems with simple bounds are considered in the present work. First, a preconditioned spectral gradient method is de ned for the case in which no simple bounds are present. This algorithm can be viewed as a quasiNewton method in which the approximate Hessians satisfy a weak secant equation.
Using Krylov Subspace and Spectral Methods for Solving Complementarity Problems in ManyBody Contact Dynamics Simulation
"... Manybody dynamics problems are expected to handle millions of unknowns when, for instance, investigating the threedimensional flow of granular material. Unfortunately, the size of the problems tractable by existing numerical solution techniques is severely limited on convergence grounds. This is t ..."
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Manybody dynamics problems are expected to handle millions of unknowns when, for instance, investigating the threedimensional flow of granular material. Unfortunately, the size of the problems tractable by existing numerical solution techniques is severely limited on convergence grounds. This is typically the case when the equations of motion embed a differential variational inequality (DVI) problem that captures contact and possibly frictional interactions between rigid and/or flexible bodies. As the size of the physical system increases, the speed and/or the quality of the numerical solution decrease. This paper describes three methods the gradient projected minimum residual (GPMINRES) method, the preconditioned spectral projected gradient with fallback (PSPGFB) method, and the Ku˘cera method that demonstrate better scalability than the projected Jacobi and GaussSeidel methods commonly used to solve contact problems that draw on a DVIbased modeling approach. Copyright c ○ 2012 John Wiley & Sons, Ltd.
IMPLICITLY PRECONDITIONED AND GLOBALIZED RESIDUAL METHOD FOR SOLVING STEADY FLUID FLOWS ∗
"... Dedicated to Víctor Pereyra on the occasion of his 70th birthday Abstract. We develop a derivativefree preconditioned residual method for solving nonlinear steady fluid flows. The new scheme is based on a variable implicit preconditioning technique associated with the globalized spectral residual m ..."
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Dedicated to Víctor Pereyra on the occasion of his 70th birthday Abstract. We develop a derivativefree preconditioned residual method for solving nonlinear steady fluid flows. The new scheme is based on a variable implicit preconditioning technique associated with the globalized spectral residual method. The new scheme is robust and allows numerical computation of the steady state of the twodimensional incompressible NavierStokes equations (NSE), which we consider here in both primary variables and streamfunctionvorticity formulations. The results are encouraging and agree with those reported in the literature.