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On a trust region method without exact Jacobian matrices
Author(s) -
Walther Andrea
Publication year - 2005
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200510015
Subject(s) - jacobian matrix and determinant , hessian matrix , constraint (computer aided design) , mathematics , trust region , factorization , mathematical optimization , algorithm , computer science , geometry , computer security , radius
Automatic differentiation (AD) provides a possibility to evaluate exact derivative information within working accuracy. Here, we present an approach for equality constrained optimization that is based essentially on AD by computing only direct sensitivities and adjoints of first and second order. Employing this information, we generate approximations of the required derivative matrices using the STR1 update instead of computing the full constraint Jacobian or the full Lagrangian Hessian at each iteration explicitly. Hence, this approach avoids the forming and factoring of the exact constraint Jacobian that is often required in each iteration step. In order to globalize this provable local convergent method, the algorithm was embedded in a trust‐region framework. We apply a composite‐step method similar to the Byrd‐Omojokun approach that is well suited for the available information from the approximated matrices. For that purpose, the normal step is given by the dogleg method whereas a generalized CG‐iteration is applied to compute the tangential step. Here, the fact that only inexact information of the constraint Jacobian is available forms the main difference to other existing algorithms, where often a factorization of the exact Jacobian is used. Numerical results are shown for an equality constrained problem of the CUTE set and a PDE‐based optimization problem. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)