An Example of Non-Convergence of Trust Region Algorithms

Abstract

It is well known that trust region algorithms have very nice convergence properties. Descent trust region algorithms can be classified into two groups. The first can be called “sufficient reduction methods” where the condition for accepting a new point is a sufficient reduction in the merit function. The other can be called “simple reduction” methods where they accept a new point as long as it reduces the merit function. In general, it can be shown that the algorithms that require sufficient reductions have strong convergence result, namely all accumulation points are stationary points. Though “simple reduction” methods have the nice property of accepting any better iterates, convegence results for these algorithms are weaker than thoes for “sufficient reduction” methods, as we are only able to show that at least one accumulation point is a stationary point.

In this paper, we construct an example to show that “simple reduction” algorithms may generate a sequence that does not converge. Instead, the sequence cycles nearly three points where only one of them is a stationary point.