Results 1  10
of
18
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 ..."
Abstract

Cited by 74 (17 self)
 Add to MetaCart
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.
Preconditioning indefinite systems in interior point methods for optimization
 Computational Optimization and Applications
, 2004
"... Abstract. Every Newton step in an interiorpoint method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today’s codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable il ..."
Abstract

Cited by 44 (13 self)
 Add to MetaCart
Abstract. Every Newton step in an interiorpoint method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today’s codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable illconditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used. Keywords: interiorpoint methods, iterative solvers, preconditioners 1.
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 ..."
Abstract

Cited by 41 (12 self)
 Add to MetaCart
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.
KNITRO: An integrated package for nonlinear optimization
 Large Scale Nonlinear Optimization, 35–59, 2006
, 2006
"... This paper describes Knitro 5.0, a Cpackage for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving largescale, smooth nonlinear programming problems ..."
Abstract

Cited by 38 (3 self)
 Add to MetaCart
This paper describes Knitro 5.0, a Cpackage for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving largescale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, least squares, and linear and quadratic programming. Various algorithmic options are available, including two interior methods and an activeset method. The package provides crossover techniques between algorithmic options as well as automatic selection of options and settings. 1
GALAHAD, a library of threadsafe Fortran 90 Packages for LargeScale Nonlinear Optimization
, 2002
"... In this paper, we describe the design of version 1.0 of GALAHAD, a library of Fortran 90 packages for largescale largescale nonlinear optimization. The library particularly addresses quadratic programming problems, containing both interior point and active set variants, as well as tools for prepro ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
In this paper, we describe the design of version 1.0 of GALAHAD, a library of Fortran 90 packages for largescale largescale nonlinear optimization. The library particularly addresses quadratic programming problems, containing both interior point and active set variants, as well as tools for preprocessing such problems prior to solution. It also contains an updated version of the venerable nonlinear programming package, LANCELOT.
A PrimalDual TrustRegion Algorithm for Minimizing a NonConvex Function Subject to General Inequality and Linear Equality Constraints
, 1999
"... A new primaldual algorithm is proposed for the minimization of nonconvex objective functions subject to general inequality and linear equality constraints. The method uses a primaldual trustregion model to ensure descent on a suitable merit function. Convergence is proved to secondorder critical ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
A new primaldual algorithm is proposed for the minimization of nonconvex objective functions subject to general inequality and linear equality constraints. The method uses a primaldual trustregion model to ensure descent on a suitable merit function. Convergence is proved to secondorder critical points from arbitrary starting points. Preliminary numerical results are presented.
An iterative workingset method for LargeScale NonConvex quadratic programming
, 2001
"... We consider a workingset method for solving largescale quadratic programming problems for which there is no requirement that the objective function be convex. The methods are iterative at two levels, one level relating to the selection of the current working set, and the second due to the method u ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
We consider a workingset method for solving largescale quadratic programming problems for which there is no requirement that the objective function be convex. The methods are iterative at two levels, one level relating to the selection of the current working set, and the second due to the method used to solve the equalityconstrained problem for this working set. A preconditioned conjugate gradient method is used for this inner iteration, with the preconditioner chosen especially to ensure feasibility of the iterates. The preconditioner is updated at the conclusion of each outer iteration to ensure that this feasibility requirement persists. The wellknown equivalence between the conjugategradient and Lanczos methods is exploited when nding directions of negative curvature. Details of an implementation  the Fortran 90 package QPA in the forthcoming GALAHAD library  are given.
Iterative Methods for IllConditioned Linear Systems From Optimization
, 1998
"... Preconditioned conjugategradient methods are proposed for solving the illconditioned linear systems which arise in penalty and barrier methods for nonlinear minimization. The preconditioners are chosen so as to isolate the dominant cause of ill conditioning. The methods are stablized using a restr ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Preconditioned conjugategradient methods are proposed for solving the illconditioned linear systems which arise in penalty and barrier methods for nonlinear minimization. The preconditioners are chosen so as to isolate the dominant cause of ill conditioning. The methods are stablized using a restricted form of iterative refinement. Numerical results illustrate the approaches considered. 1 Email : n.gould@rl.ac.uk 2 Current reports available from "http://www.rl.ac.uk/departments/ccd/numerical/reports/reports.html". Department for Computation and Information Atlas Centre Rutherford Appleton Laboratory Oxfordshire OX11 0QX August 26, 1998. 1 INTRODUCTION 1 1 Introduction Let A and H be, respectively, fullrank m by n (m n) and symmetric n by n real matrices. Suppose furthermore that any nonzero coefficients in this data are modest, that is the data is O(1). (1) We consider the iterative solution of the linear system (H +A T D \Gamma1 A)x = b (1.1) where b is modest an...
An activeset algorithm for nonlinear programming using linear programming and equality constrained subproblems
, 2002
"... 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 [9]. The step computation is performed in two stages. In the rst stage a linear program is solved to estimate the active set ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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 [9]. The step computation is performed in two stages. In the rst 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 atthesolution of the linear program. The EQP incorporates a trustregion constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr [1] test set.
A Sequential Quadratic Programming Algorithm with an Additional Equality Constrained Phase
, 2008
"... A sequential quadratic programming (SQP) method is presented that aims to overcome some of the drawbacks of contemporary SQP methods. It avoids the difficulties associated with indefinite quadratic programming subproblems by defining this subproblem to be always convex. The novel feature of the appr ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
A sequential quadratic programming (SQP) method is presented that aims to overcome some of the drawbacks of contemporary SQP methods. It avoids the difficulties associated with indefinite quadratic programming subproblems by defining this subproblem to be always convex. The novel feature of the approach is the addition of an equality constrained phase that promotes fast convergence and improves performance in the presence of ill conditioning. This equality constrained phase uses exact second order information and can be implemented using either a direct solve or an iterative method. The paper studies the global and local convergence properties of the new algorithm and presents a set of numerical experiments to illustrate its practical performance.