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Abstract.A subspace adaptation of the Coleman-Li trust region and interior method is proposed for solving large-scale bound-constrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its full-space version.Computational performance on various large-scale test problems are reported; advantages of our approach are demonstrated. Our experience indicates our proposed method represents an efficient way to solve large-scale bound-constrained minimization problems.

Given a mapping with a sparse Jacobian matrix, we investigate the problem of minimizing the number of function evaluations needed to estimate the Jacobian matrix by differences. We show that this problem can be attacked as a graph coloring problem and that this approach leads to very efficient algorithms. The behavior of these algorithms is studied and, in particular, we prove that two of the algorithms are optimal for band graphs. We also present numerical evidence which indicates that these two algorithms are nearly optimal on practical problems.

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