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TrustRegion InteriorPoint Algorithms For Minimization Problems With Simple Bounds
 SIAM J. Control and Optimization
, 1995
"... . Two trustregion interiorpoint algorithms for the solution of minimization problems with simple bounds are analyzed and tested. The algorithms scale the local model in a way similar to Coleman and Li [1]. The first algorithm is more usual in that the trust region and the local quadratic model a ..."
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Cited by 48 (19 self)
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. Two trustregion interiorpoint algorithms for the solution of minimization problems with simple bounds are analyzed and tested. The algorithms scale the local model in a way similar to Coleman and Li [1]. The first algorithm is more usual in that the trust region and the local quadratic model are consistently scaled. The second algorithm proposed here uses an unscaled trust region. A global convergence result for these algorithms is given and dogleg and conjugategradient algorithms to compute trial steps are introduced. Some numerical examples that show the advantages of the second algorithm are presented. Keywords. trustregion methods, interiorpoint algorithms, DikinKarmarkar ellipsoid, Coleman and Li affine scaling, simple bounds. AMS subject classification. 49M37, 90C20, 90C30 1. Introduction. In this note we consider the boxconstrained minimization problem minimize f(x) subject to a x b; (1) where x 2 IR n , a 2 (IR [ f\Gamma1g) n , b 2 (IR [ f+1g) n and...
An Algorithm for Nonlinear Optimization Using Linear Programming and Equality Constrained Subproblems
, 2003
"... This paper describes an activeset algorithm for largescale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the activ ..."
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Cited by 41 (12 self)
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This paper describes an activeset algorithm for largescale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the ` 1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program.
Recent Progress in Unconstrained Nonlinear Optimization Without Derivatives
 MATHEMATICAL PROGRAMMING
, 1997
"... We present an introduction to a new class of derivative free methods for unconstrained optimization. We start by discussing the motivation for such methods and why they are in high demand by practitioners. We then review the past developments in this field, before introducing the features that ch ..."
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Cited by 37 (2 self)
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We present an introduction to a new class of derivative free methods for unconstrained optimization. We start by discussing the motivation for such methods and why they are in high demand by practitioners. We then review the past developments in this field, before introducing the features that characterize the newer algorithms. In the context of a trust region framework, we focus on techniques that ensure a suitable "geometric quality" of the considered models. We then outline the class of algorithms based on these techniques, as well as their respective merits. We finally conclude the paper with a discussion of open questions and perspectives.
TrustRegion InteriorPoint SQP Algorithms For A Class Of Nonlinear Programming Problems
 SIAM J. CONTROL OPTIM
, 1997
"... In this paper a family of trustregion interiorpoint SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal co ..."
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Cited by 35 (8 self)
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In this paper a family of trustregion interiorpoint SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed for a different class of problems by Coleman and Li and they exploit trustregion techniques for equalityconstrained optimizatio...
Fast Recursive SQP Methods for LargeScale Optimal Control Problems
, 1995
"... Direct boundary value problem methods in combination with SQP iteration have proved to be very successful in solving nonlinear optimal control problems. Such methods use parameterized control functions, discretize the state differential equations by, e.g., multiple shooting or collocation, and treat ..."
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Cited by 23 (8 self)
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Direct boundary value problem methods in combination with SQP iteration have proved to be very successful in solving nonlinear optimal control problems. Such methods use parameterized control functions, discretize the state differential equations by, e.g., multiple shooting or collocation, and treat the discretized boundary value problem as an equality constraint in a large, nonlinear, constrained optimization problem. In reallife applications several thousand variables may appear in the NLP. Solution by standard techniques is therefore impractical. This dissertation develops a general concept for a class of structured direct SQP methods based on a decoupling strategy. A careful choice of the discretization reveals an inherent multistage block structure of the QP subproblems. We present a recursive solution algorithm for the associated KKT systems which makes full use of this sparse structure, and propose a structurepreserving primaldual interior point method for treating the genera...
Solving the trustregion subproblem using the Lanczos method
, 1997
"... The approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods. When the number of variables is large, the most widelyused strategy is to trace the path of conjugate gradient iterates either to convergence or ..."
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Cited by 21 (1 self)
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The approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods. When the number of variables is large, the most widelyused strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trustregion boundary. In this paper, we investigate ways of continuing the process once the boundary has been encountered. The key is to observe that the trustregion problem within the currently generated Krylov subspace has very special structure which enables it to be solved very efficiently. We compare the new strategy with existing methods. The resulting software package is available as HSL VF05 within the Harwell Subroutine Library. 1 Department for Computation and Information, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU Email : n.gould@rl.ac.uk 2 Current reports available by anonymous ftp from joyousgard.cc.rl.ac.uk (internet ...
Combining Trust Region and Line Search Techniques
"... We propose an algorithm for nonlinear optimization that employs both trust region techniques and line searches. Unlike traditional trust region methods, our algorithm does not resolve the subproblem if the trial step results in an increase in the objective function, but instead performs a backtr ..."
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Cited by 17 (3 self)
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We propose an algorithm for nonlinear optimization that employs both trust region techniques and line searches. Unlike traditional trust region methods, our algorithm does not resolve the subproblem if the trial step results in an increase in the objective function, but instead performs a backtracking line search from the failed point. Backtracking can be done along a straight line or along a curved path. We show that the new algorithm preserves the strong convergence properties of trust region methods. Numerical results are also presented.
An overview of unconstrained optimization
 Online]. Available: citeseer.ist.psu.edu/fletcher93overview.html 150
, 1993
"... bundle filter method for nonsmooth nonlinear ..."
Global convergence of general derivativefree trustregion algorithms to first and second order critical points
, 2006
"... Abstract. In this paper we prove global convergence for first and secondorder stationary points of a class of derivativefree trustregion methods for unconstrained optimization. These methods are based on the sequential minimization of quadratic (or linear) models built from evaluating the objecti ..."
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Cited by 13 (2 self)
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Abstract. In this paper we prove global convergence for first and secondorder stationary points of a class of derivativefree trustregion methods for unconstrained optimization. These methods are based on the sequential minimization of quadratic (or linear) models built from evaluating the objective function at sample sets. The derivativefree models are required to satisfy Taylortype bounds but, apart from that, the analysis is independent of the sampling techniques. A number of new issues are addressed, including global convergence when acceptance of iterates is based on simple decrease of the objective function, trustregion radius maintenance at the criticality step, and global convergence for secondorder critical points. Key words. TrustRegion Methods, DerivativeFree Optimization, Nonlinear Optimization,
A LargeScale TrustRegion Approach to the Regularization of Discrete IllPosed Problems
 RICE UNIVERSITY
, 1998
"... We consider the problem of computing the solution of largescale discrete illposed problems when there is noise in the data. These problems arise in important areas such as seismic inversion, medical imaging and signal processing. We pose the problem as a quadratically constrained least squares pro ..."
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Cited by 12 (4 self)
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We consider the problem of computing the solution of largescale discrete illposed problems when there is noise in the data. These problems arise in important areas such as seismic inversion, medical imaging and signal processing. We pose the problem as a quadratically constrained least squares problem and develop a method for the solution of such problem. Our method does not require factorization of the coefficient matrix, it has very low storage requirements and handles the high degree of singularities arising in discrete illposed problems. We present numerical results on test problems and an application of the method to a practical problem with real data.