Abstract

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 di®erence 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.