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43
A trust region method based on interior point techniques for nonlinear programming
 Mathematical Programming
, 1996
"... Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direc ..."
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Cited by 105 (18 self)
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Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primaldual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Key words: constrained optimization, interior point method, largescale optimization, nonlinear programming, primal method, primaldual method, SQP iteration, barrier method, trust region method.
An interior point algorithm for large scale nonlinear programming
 SIAM Journal on Optimization
, 1999
"... The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primaldual versions of t ..."
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Cited by 74 (17 self)
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The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primaldual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests. Key words: constrained optimization, interior point method, largescale optimization, nonlinear programming, primal method, primaldual method, successive quadratic programming, trust region method.
On affine invariant clustering and automatic cast listing in movies
 In Proc. ECCV
, 2002
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Global Convergence of a Class of Trust Region Algorithms for Optimization Using Inexact Projections on Convex Constraints
, 1995
"... A class of trust region based algorithms is presented for the solution of nonlinear optimization problems with a convex feasible set. At variance with previously published analysis of this type, the theory presented allows for the use of general norms. Furthermore, the proposed algorithms do not r ..."
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Cited by 54 (4 self)
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A class of trust region based algorithms is presented for the solution of nonlinear optimization problems with a convex feasible set. At variance with previously published analysis of this type, the theory presented allows for the use of general norms. Furthermore, the proposed algorithms do not require the explicit computation of the projected gradient, and can therefore be adapted to cases where the projection onto the feasible domain may be expensive to calculate. Strong global convergence results are derived for the class. It is also shown that the set of linear and nonlinear constraints that are binding at the solution are identified by the algorithms of the class in a finite number of iterations.
A New Trust Region Algorithm For Equality Constrained Optimization
, 1995
"... . We present a new trust region algorithm for solving nonlinear equality constrained optimization problems. At each iterate a change of variables is performed to improve the ability of the algorithm to follow the constraint level sets. The algorithm employs L 2 penalty functions for obtaining global ..."
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Cited by 51 (7 self)
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. We present a new trust region algorithm for solving nonlinear equality constrained optimization problems. At each iterate a change of variables is performed to improve the ability of the algorithm to follow the constraint level sets. The algorithm employs L 2 penalty functions for obtaining global convergence. Under certain assumptions we prove that this algorithm globally converges to a point satisfying the second order necessary optimality conditions; the local convergence rate is quadratic. Results of preliminary numerical experiments are presented. 1. Introduction. We consider the equality constrained optimization problem minimize f(x) subject to c(x) = 0 (1:1) where x 2 ! n and f : ! n ! !, and c : ! n ! ! m are smooth nonlinear functions. Problem (1.1) is often solved by successive quadratic programming (SQP) methods. At a current point x k 2 ! n , SQP methods determine a search direction d k by solving a quadratic programming problem minimize rf(x k ) T d + 1 2 ...
A Global Convergence Theory for General TrustRegionBased Algorithms for Equality Constrained Optimization
 SIAM Journal on Optimization
, 1992
"... This work presents a global convergence theory for a broad class of trustregion algorithms for the smooth nonlinear progro.mmln S problem with equality constraints. The main result generalizes Powell's 1975 result for unconstrained trustregion algorithms. ..."
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Cited by 44 (10 self)
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This work presents a global convergence theory for a broad class of trustregion algorithms for the smooth nonlinear progro.mmln S problem with equality constraints. The main result generalizes Powell's 1975 result for unconstrained trustregion algorithms.
On the implementation of an algorithm for largescale equality constrained optimization
 SIAM Journal on Optimization
, 1998
"... Abstract. This paper describes a software implementation of Byrd and Omojokun’s trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques ..."
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Cited by 40 (11 self)
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Abstract. This paper describes a software implementation of Byrd and Omojokun’s trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques for solving the subproblems occurring in the algorithm. Second derivative information can be used, but when it is not available, limited memory quasiNewton approximations are made. The performance of the code is studied using a set of difficult test problems from the CUTE collection.
Trust Region Algorithms For Constrained Optimization
 Math. Prog
, 1990
"... We review the main techniques used in trust region algorithms for nonlinear constrained optimization. 1. Trust Region Idea Constrained optimization is to minimize a function subject to finitely many algebraic equation and inequality conditions. It has the following form min x2! n f(x) (1.1) subj ..."
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Cited by 26 (6 self)
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We review the main techniques used in trust region algorithms for nonlinear constrained optimization. 1. Trust Region Idea Constrained optimization is to minimize a function subject to finitely many algebraic equation and inequality conditions. It has the following form min x2! n f(x) (1.1) subject to c i (x) = 0; i = 1; 2; : : : ; m e ; (1.2) c i (x) 0; i = m e + 1; : : : ; m; (1.3) where f(x) and c i (x) (i = 1; : : : ; m) are real functions defined in ! n , and m m e are two nonnegative integers. Numerical methods for nonlinear optimization problems can be grouped as two types. One are line search methods and the other are trust region algorithms. Line search algorithms at each iteration use a direction to carry a line search. The direction is called the search direction, which is normally computed by solving a subproblem that approximates the original problem near the current iterate. A line search means to search for a new point along the search direction. For example, ...
Global Convergence of TrustRegion SQPFilter Algorithms for General Nonlinear Programming
, 1999
"... Global convergence to firstorder critical points is proved for two trustregion SQPfilter algorithms of the type introduced by Fletcher and Leyffer (1997). The algorithms allow for an approximate solution of the quadratic subproblem and incorporate the safeguarding tests described in Fletcher, Ley ..."
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Cited by 26 (2 self)
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Global convergence to firstorder critical points is proved for two trustregion SQPfilter algorithms of the type introduced by Fletcher and Leyffer (1997). The algorithms allow for an approximate solution of the quadratic subproblem and incorporate the safeguarding tests described in Fletcher, Leyffer and Toint (1998). The first algorithm decomposes the step into its normal and tangential components, while the second replaces this decomposition by a stronger condition on the associated model decrease. 1 Department of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, EU. Email : fletcher@mcs.dundee.ac.uk, sleyffer@mcs.dundee.ac.uk 2 Current reports available from "http://www.mcs.dundee.ac.uk:8080/~dfg/Narep.html". 3 Computational Science and Engineering Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU. Email : n.gould@rl.ac.uk 4 Current reports available from "http://www.numerical.rl.ac.uk/reports/reports.html". 5 Department ...