Journal of Inverse and Ill-posed Problems
Editor-in-Chief: Kabanikhin, Sergey I.
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On the ill-posedness of the trust region subproblem
- Laboraroire Informatique Théorique & Appliquée (LITA EA 3097), UFR MIM, Université de Metz, Ile du Saulcy, F57045 Metz Cedex 01, France. E-mail:
- Laboratoire de Mathématique (LMI), Institut National des Sciences Appliquée (INSA) de Rouen, BP 8, F76131 Mont Saint Aignan, France. E-mails: ,
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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.