Solving Trust Region Problem in Large Scale Optimization

Abstract

This paper presents a new method for solving the basic problem in the "model-trust region" approach to large scale minimization: Compute a vector $x$ such that $1/2 x^THx +c^Tx $ = min, subject to the constraint $\| x \|_2 ≤a$. The method is a combination of the CG method and a projection and contraction (PC) method. The first (CG) method with $x_0 = 0$ as the starting point either directly offers a solution of the problem, or — as soon as the norm of the iteration is greater than $a$, — it gives a suitable starting point and a favourable choice of a crucial scaling parameter in the second (PC) method. Some numerical examples are given, which indicate that the method is applicable.