QN Optimization with Hessian Sample

Abstract: 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 $Y=\nabla² f(x)\,S$ whenever the gradient is evaluated. Block QN algorithms then update approximate inverse Hessians, $H_k \approx \nabla² f(x_k)$, 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 $S$.