Computing a Trust Region Step

An algorithm is proposed for the problem of minimizing a quadratic function subject to an ellipsoidal constraint which is guaranteed to produce a nearly optimal solution in a finite number of iterations. A robust and efficient algorithm for this problem is required to compute the step between iterates in trust region methods for optimization problems. We also consider the use of our algorithm in a trust region Newton's method. In particular, we prove that under reasonable assumptions the sequence (X/sub k/) generated by Newton's method has a limit point X* which satisfies the first and second order necessary conditions for a minimizer of the objective function f. Numerical results for GQTPAR, which is a Fortran implementation of our algorithm, show that GQTPAR is quite successful in a trust region method. In our tests a call to GQTPAR only required 1.6 iterations on the average.