A new active-set method for smooth box-constrained minimization is introduced. The algorithm combines an unconstrained method, including a new line-search 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: Box-constrained minimization, numerical methods, activeset strategies, Spectral Projected Gradient. 1

Abstract. Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems in which the constraints are only of the lower-level type. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of a comparison against Ipopt and Lancelot B. The resolution of location problems in which many constraints of the lower-level set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time. The codes are free for download in www.ime.usp.br/∼egbirgin/tango/

A practical algorithm for box-constrained optimization is introduced. The algorithm combines an active-set 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

Augmented Lagrangian algorithms are very popular tools for solving nonlinear programming problems. At each outer iteration of these methods a simpler optimization problem is solved, for which ecient algorithms can be used, especially when the problems are large. The most famous Augmented Lagrangian algorithm for minimization with inequality constraints is known as Powell-Hestenes-Rockafellar (PHR) method. The main drawback of PHR is that the objective function of the subproblems is not twice continuously dierentiable. This is the main motivation for the introduction of many alternative Augmented Lagrangian methods.

Abstract. An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventually reduces to unconstrained optimization without restarts. Similarly, for a degenerate stationary point, where the strong secondorder sufficient optimality condition holds, the algorithm eventually reduces to unconstrained optimization without restarts. A specific implementation of the ASA is given which exploits the recently developed cyclic Barzilai–Borwein (CBB) algorithm for the gradient projection step and the recently developed conjugate gradient algorithm CG DESCENT for unconstrained optimization. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK-2 test problem libraries. Key words. nonmonotone gradient projection, box constrained optimization, active set algorithm,

BOX-QUACAN is a trust-region box-constraint optimization software developed at the Applied Mathematics Department of the University of Campinas. During the last five years, it has been used for solving many practical and academic problems with box constraints and it has been incorporated as sub-algorithm of Augmented Lagrangian methods for minimization with equality constraints and bounds. In this paper it is described its use in connection with Augmented Lagrangian algorithms where inequality constraints are handled without the addition of slack variables. Numerical experiments comparing a modified exponential Lagrangian method and the most classical Augmented Lagrangian are presented. Institute of Mathematics, University of Campinas, CP 6065, 13081-970 Campinas SP, Brazil. This work was supported by PRONEX, FAPESP (grant 90-3724-6), FINEP, CNPq, FAEP-UNICAMP. 1 1 Introduction Box constrained optimization is a well developed area of numerical analysis. It consists on the m...

Communicated by C. T. Leondes 1This work was supported by PRONEX-CNPq/FAPERJ Grant E-26/171.164/2003- APQ1, FAPESP Grants 03/09169-6 and 01/04597-4, and CNPq. The authors are indebted to Juliano B. Francisco and Yalcin Kaya for their careful reading of the first draft of this paper.

Summary. The contribution deals with the numerical solving of contact problems with Coulomb friction for 3D bodies. A variant of the FETI based domain decomposition method is used. Numerical experiments illustrate the efficiency of our algorithm. 1

this paper, we show that the computational cost of the contact shape optimization may be essentially reduced by the application of a domain decomposition method to the solution of the state variational inequality that describes the equilibrium of a system of elastic bodies. In particular, we describe an algorithm for the minimization of a compliance of one body in a coercive system of bodies during their mutual contacts. After discretization by the finite element method, the algorithm uses a feasible directions method for minimization of the cost functional.

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