caa a22 4500 378879820 CHVBK 20180305123429.0 cr unu---uuuuu 161128e20031215xx s 000 0 eng 10.1515/156939403322759642 doi (NATIONALLICENCE)gruyter-10.1515/156939403322759642 On the ill-posedness of the trust region subproblem [Elektronische Daten] [Le Thi Hoai An, Pham Dinh Tao, Dinh Nho Hào] The trust region subproblem plays an important role in optimization and numerical analysis. Many researchers even use it in regularizing ill-posed problems. It appears that the trust region subproblem is ill-posed: the set of solutions is unstable with respect to the data in the functional to be minimized, that is a small error in the functional to be minimized might cause large errors in the set of solutions. The aim of the paper is to study the ill-posed nature of the problem and to suggest methods to overcome the ill-posedness. The methods are mainly based on Tikhonov regularization with the generalized discrepancy principle suggested by Goncharskii, Leonov, and Yagola and the difference of convex functions algorithm (DCA) recently developed by Pham Dinh Tao and Le Thi Hoai An. The open problem of Tikhonov regularization methods for non-linear ill-posed problems how to globally solve non-linear (in general non-convex) optimization problems occurred from them is completely answered for the trust region subproblem by DCA. Several test numerical examples are outlined. Copyright 2003, Walter de Gruyter An Le Thi Hoai Laboraroire Informatique Théorique & Appliquée (LITA EA 3097), UFR MIM, Université de Metz, Ile du Saulcy, F57045 Metz Cedex 01, France. E-mail: lethi@sciences.univ-metz.fr aut Tao Pham Dinh Laboratoire de Mathématique (LMI), Institut National des Sciences Appliquée (INSA) de Rouen, BP 8, F76131 Mont Saint Aignan, France. E-mails: lethi@insa-rouen.fr, pham@insarouen.fr aut Hào Dinh Nho Hanoi Institute of Mathematics, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam and Vrije Universiteit Brussel (VUB), Department of Electronics and Information Processing (ETRO), Pleinlaan 2, 1050 Brussels, Belgium. E-mails: hao@math.ac.vn, hao@etro.vub.ac.be. aut Journal of Inverse and Ill-posed Problems Walter de Gruyter 11/6(2003-12-15), 545-577 0928-0219 11:6<545 2003 11 jiip https://doi.org/10.1515/156939403322759642 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.1515/156939403322759642 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- An Le Thi Hoai Laboraroire Informatique Théorique & Appliquée (LITA EA 3097), UFR MIM, Université de Metz, Ile du Saulcy, F57045 Metz Cedex 01, France. E-mail: lethi@sciences.univ-metz.fr aut NATIONALLICENCE 700 1- Tao Pham Dinh Laboratoire de Mathématique (LMI), Institut National des Sciences Appliquée (INSA) de Rouen, BP 8, F76131 Mont Saint Aignan, France. E-mails: lethi@insa-rouen.fr, pham@insarouen.fr aut NATIONALLICENCE 700 1- Hào Dinh Nho Hanoi Institute of Mathematics, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam and Vrije Universiteit Brussel (VUB), Department of Electronics and Information Processing (ETRO), Pleinlaan 2, 1050 Brussels, Belgium. E-mails: hao@math.ac.vn, hao@etro.vub.ac.be aut NATIONALLICENCE 773 0- Journal of Inverse and Ill-posed Problems Walter de Gruyter 11/6(2003-12-15), 545-577 0928-0219 11:6<545 2003 11 jiip CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter