In this paper, we first discuss how the nearly exact (NE) method proposed by Moré and Sorensen  for solving trust region (TR) subproblems can be modified to solve large-scale "low-rank" TR subproblems efficiently. Our modified algorithm completely avoids computation of Cholesky factorizations by instead relying primarily on the Sherman-Morrison-Woodbury formula for computing inverses of "diagonal plus low-rank" type matrices. We also implement a specific version of the modified log-barrier (MLB) algorithm proposed by Polyak  where the generated log-barrier subproblems are solved by a trust region method. The corresponding direction finding TR subproblems are of the low-rank type and are then solved by our modified NE method. We finally discuss the computational results of our implementation of the MLB method and its comparison with a version of LANCELOT  based on a collection extracted from CUTEr  of nonlinear programming problems with simple bound constraints.
- Large-scale optimization
- Limited-memory BFGS method
- Nearly exact method
- Sherman-Morrison-Woodbury formula
- Trust region method