A matrix-free approach to build band preconditioners for large-scale bound-constrained optimization

We propose a procedure for building symmetric positive definite band preconditioners for large-scale symmetric, possibly indefinite, linear systems, when the coefficient matrix is not explicitly available, but matrix–vector products involving it can be computed. We focus on linear systems arising in Newton-type iterations within matrix-free versions of projected methods for bound-constrained nonlinear optimization. In this case, the structure and the size of the matrix may significantly change in subsequent iterations, and preconditioner updating algorithms that exploit information from previous steps cannot be easily applied. Our procedure is based on a recursive approach that incrementally improves the quality of the preconditioner, while requiring a modest number of matrix–vector products. A strategy for dynamically choosing the bandwidth of the preconditioners is also presented. Numerical results are provided, showing the performance of our preconditioning technique within a trust-region Newton method for bound-constrained optimization.