Computing a Trust Region Step for a Penalty Function

The problem of minimizing a quadratic function subject to an ellipsoidal constraint when the matrix involved is the Hessian of a quadratic penalty function (i.e., a function of the form p(x) = f(x) + (1/2μ )c(x)T c(x)) is considered. Most applications of penalty functions require $p(x)$ to be minimized for values of μ decreasing to zero. In general, as μ tends to zero the nature of finite precision arithmetic causes a considerable loss of information about the null space of the constraint gradients when ∇² p(x) is formed. This loss of information renders ordinary trust region Newton’s methods unstable and degrades the accuracy of the solution to the trust region problem. The algorithm of Moré and Sorensen [SIAM J. Sci. Statist. Comput., 4 (1983), pp. 553–572] is modified so as to be more stable and less sensitive to the nature of finite precision arithmetic in this situation. Numerical experiments clearly demonstrate the stability of the proposed algorithm.