# PNKH-B: A Projected Newton-Krylov Method for Large-Scale Bound-Constrained Optimization

We present PNKH-B, a projected Newton-Krylov method with a low-rank approximated Hessian metric for approximately solving large-scale optimization problems with bound constraints. PNKH-B is geared toward situations in which function and gradient evaluations are expensive, and the (approximate) Hessian is only available through matrix-vector products. This is commonly the case in large-scale parameter estimation, machine learning, and image processing. In each iteration, PNKH-B generates a low-rank approximation of the (approximate) Hessian using Lanczos tridiagonalization and then solves a quadratic projection problem to update the iterate. The key idea is to compute the projection with respect to the norm defined by the low-rank approximation. Hence, PNKH-B can be viewed as a projected variable metric method. We present an interior point method to solve the quadratic projection problem efficiently. Since the interior point method effectively exploits the low-rank structure, its computational cost only scales linearly with respect to the number of variables, and it only adds negligible computational time. We also experiment with variants of PNKH-B that incorporate estimates of the active set into the Hessian approximation. We prove the global convergence to a stationary point under standard assumptions. Using three numerical experiments motivated by parameter estimation, machine learning, and image reconstruction, we show that the consistent use of the Hessian metric in PNKH-B leads to fast convergence, particularly in the first few iterations. We provide our MATLAB implementation at https://github.com/EmoryMLIP/PNKH-B.

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