A Subspace Minimization Method for the Trust-Region Step

We consider methods for large-scale unconstrained minimization based on finding an approximate minimizer of a quadratic function subject to a two-norm trust-region constraint. The Steihaug-Toint method uses the conjugate-gradient method to minimize the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constraint boundary. However, if the conjugate-gradient method is used with a preconditioner, the Steihaug-Toint method requires that the trust-region norm be defined in terms of the preconditioning matrix. If a different preconditioner is used for each subproblem, the shape of the trust-region can change substantially from one subproblem to the next, which invalidates many of the assumptions on which standard methods for adjusting the trust-region radius are based. In this paper we propose a method that allows the trust-region norm to be defined independently of the preconditioner. The method solves the inequality constrained trust-region subproblem over a sequence of evolving low-dimensional subspaces. Each subspace includes an accelerator direction defined by a regularized Newton method for satisfying the optimality conditions of a primal-dual interior method. A crucial property of this direction is that it can be computed by applying the preconditioned conjugate-gradient method to a positive-definite system in both the primal and dual variables of the trust-region subproblem. Numerical experiments on problems from the CUTEr test collection indicate that the method can require significantly fewer function evaluations than other methods. In addition, experiments with general-purpose preconditioners show that it is possible to significantly reduce the number of matrix-vector products relative to those required without preconditioning.