Two Interior Trust Region Methods For Large-scale Nonlinear Optimization Subject To Bounds

Two interior trust region methods for large-scale nonlinear minimization subject to simple bounds are proposed in this paper. The bound-constrained problem is transformed into a quadratic trust region model by affine scaling transformation based on the first-order and second-order optimal conditions of the original problem. Then the LMTI approach and the STI approach are proposed, which execute a subspace implementation during the course of constructing and solving the quadratic model respectively. For the LMTI approach, the limited memory BFGS formula, which involves a compact representation of the BFGS matrix is used to update the approximation of Hessian matrix. While for the STI approach we construct a two-dimensional subspace and then solve the trust region subproblem on it directly. During the iterations the trial step is truncated when necessary in order to keep the strict feasibility of the iteration points.It is shown that the LMTI approach and the STI approach have global convergence properties under some reasonable assumptions and will achieve local quadratic convergence rate under some even stronger conditions. So the convergence properties of the subspace trust region methods are as strong as those of its full-space version.Computational results of the approaches show that they are suitable for large-scale nonlinear bound-constrained problems and the number of iterations will not increase rapidly with the scale-up of the problems. The LMTI approach outperforms the STI approach on the whole. For most of the tests, it can converge to the optimal solution in fewer iterations. Both approaches can be competitive with the L-BFGS-B method on some test problems, which proves their validity.