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66
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 147 (19 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.
CUTEr (and SifDec), a constrained and unconstrained testing environment, revisited
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 2001
"... The initial release of CUTE, a widely used testing environment for optimization software was described in [2]. The latest version, now known as CUTEr is presented. New features include reorganisation of the environment to allow simultaneous multiplatform installation, new tools for, and interface ..."
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Cited by 85 (8 self)
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The initial release of CUTE, a widely used testing environment for optimization software was described in [2]. The latest version, now known as CUTEr is presented. New features include reorganisation of the environment to allow simultaneous multiplatform installation, new tools for, and interfaces to, optimization packages, and a considerably simplified and entirely automated installation procedure for unix systems. The SIF decoder, which used to be a part of CUTE, has become a separate tool, easily callable by various packages. It features simple extensions to the SIF test problem format and the generation of files suited to automatic differentiation packages.
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 69 (7 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.
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 are c ..."
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Cited by 55 (18 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.
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 51 (11 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 46 (12 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.
Strong Duality in Nonconvex Quadratic Optimization with Two Quadratic Constraints
 SIAM JOURNAL ON OPTIMIZATION
, 2006
"... We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection betwe ..."
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Cited by 40 (9 self)
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We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection between the image of the real and complex spaces under a quadratic mapping, which together with the results in the complex case lead to a condition that ensures strong duality in the real setting. Preliminary numerical simulations suggest that for random instances of the extended trust region subproblem, the sufficient condition is satisfied with a high probability. Furthermore, we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems. Finally, we discuss an application of our results to robust least squares problems.
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 37 (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, ...
Approximation Algorithms for Quadratic Programming
, 1998
"... We consider the problem of approximating the global minimum of a general quadratic program (QP) with n variables subject to m ellipsoidal constraints. For m = 1, we rigorously show that an fflminimizer, where error ffl 2 (0; 1), can be obtained in polynomial time, meaning that the number of arithme ..."
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Cited by 31 (5 self)
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We consider the problem of approximating the global minimum of a general quadratic program (QP) with n variables subject to m ellipsoidal constraints. For m = 1, we rigorously show that an fflminimizer, where error ffl 2 (0; 1), can be obtained in polynomial time, meaning that the number of arithmetic operations is a polynomial in n, m, and log(1=ffl). For m 2, we present a polynomialtime (1 \Gamma 1 m 2 )approximation algorithm as well as a semidefinite programming relaxation for this problem. In addition, we present approximation algorithms for solving QP under the box constraints and the assignment polytope constraints. Key words. Quadratic programming, global minimizer, polynomialtime approximation algorithm The work of the first author was supported by the Australian Research Council; the second author was supported in part by the Department of Management Sciences of the University of Iowa where he performed this research during a research leave, and by the Natural Scien...