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 lowdimensional 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 conjugategradient method to a positive-definite system in both the primal and dual variables of the trustregion 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. [PUBLICATION ABSTRACT]