An affine-scaling interior-point CBB method for box-constrained optimization
type: Item
instance of: scholarly article
title: An affine-scaling interior-point CBB method for box-constrained optimization
zbMATH Open document ID: 1168.90007
author: William Hager
author: Bernard A. Mair
author: Hongchao Zhang
published in: Mathematical Programming. Series A. Series B
publication date: 2009-04-24T00:00:00Z
review text: An affine-scaling interior-point algorithm for box-constrained nonlinear optimization problems is developed. The k-th direction in the algorithm is a quasi-Newton direction for the first order optimality (KKT) conditions for the problem, where in regard to large systems as they occur, for example, in the context of image reconstruction for positron emission tomography (PET), the current Hessian of the objective function is replaced by a suitable multiple of the unit matrix. The respective factor for the unit matrix is computed by a cyclic modification of a rule which is used in the Barzilai-Borwein (BB) quasi-Newton method. In order to relate to further original features of BB type methods which typically do not reduce the value of the cost function in a monotonic way, the method is connected with a nonmonotone line search procedure. The authors prove global convergence of the method as well as local r-linear convergence to a local minimizer which is nondegenerate and satisfies the second order sufficiency optimality conditions. Numerical experiments demonstrate that the speed of convergence of the method is relatively independent of the conditioning of the problem. They moreover indicate that the new method is more efficient than the earlier developed conjugate gradient based active set algorithm ASA if a not too small error tolerance is chosen, but that it is inferior to the latter algorithm if many iterations are needed and the error tolerance is tiny respectively.
reviewed by: Rembert Reemtsen
Mathematics Subject Classification ID: 90C06
Mathematics Subject Classification ID: 90C26
Mathematics Subject Classification ID: 65Y20
zbMATH DE Number: 5547227
zbMATH Keywords: Interior-point
zbMATH Keywords: affine-scaling
zbMATH Keywords: cyclic Barzilai-Borwein methods
zbMATH Keywords: image reconstruction
zbMATH Keywords: PET
zbMATH Keywords: global convergence
zbMATH Keywords: local convergence
MaRDI profile type: Publication
full work available at URL: s10107-007-0199-0
OpenAlex ID: W2125762761
cites work: On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method
cites work: Two-Point Step Size Gradient Methods
cites work: Estimation of the optimal constants and the thickness of thin films using unconstrained optimization
cites work: Nonmonotone Spectral Projected Gradient Methods on Convex Sets
cites work: On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds
cites work: An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds
cites work: A trust region and affine scaling interior point method for nonconvex minimization with linear inequality constraints
cites work: Alternate step gradient method*
cites work: Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming
cites work: The cyclic Barzilai-–Borwein method for unconstrained optimization
cites work: Adaptive two-point stepsize gradient algorithm
cites work: Q5583564
cites work: Box Constrained Quadratic Programming with Proportioning and Projections
cites work: Q5702604
cites work: Gradient Method with Retards and Generalizations
cites work: A New Active Set Algorithm for Box Constrained Optimization
cites work: Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumption
cites work: On affine-scaling interior-point Newton methods for nonlinear minimization with bound constraints
cites work: The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem
cites work: A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization
cites work: Q5393849
DOI: 10.1007/S10107-007-0199-0
description: scientific article; zbMATH DE number 5547227