  This article explores how to effectively incorporate curvature information
generated using SIMD-parallel forward-mode Algorithmic Differentiation (AD)
into unconstrained Quasi-Newton (QN) minimization of a smooth objective
function, $f$. Specifically, forward-mode AD can be used to generate block
Hessian samples $Y=\nabla^2 f(x)\,S$ whenever the gradient is evaluated. Block
QN algorithms then update approximate inverse Hessians, $H_k \approx \nabla^2
f(x_k)$, with these Hessian samples. Whereas standard line-search based BFGS
algorithms carefully filter and correct secant-based approximate curvature
information to maintain positive definite approximations, our algorithms
directly incorporate Hessian samples to update indefinite inverse Hessian
approximations without filtering. The sampled directions supplement the
standard QN two-dimensional trust-region sub-problem to generate a moderate
dimensional subproblem which can exploit negative curvature. The resulting
quadratically-constrained quadratic program is solved accurately with a
generalized eigenvalue algorithm and the step advanced using standard trust
region step acceptance and radius adjustments. The article aims to avoid serial
bottlenecks, exploit accurate positive and negative curvature information, and
conduct a preliminary evaluation of selection strategies for $S$.
