Objective acceleration for unconstrained optimization

Acceleration schemes can dramatically improve existing optimization procedures. In most of the work on these schemes, such as nonlinear generalized minimal residual (N‐GMRES), acceleration is based on minimizing the ?₂ norm of some target on subspaces of ${\mathbb{R}}^n$. There are many numerical examples that show how accelerating general‐purpose and domain‐specific optimizers with N‐GMRES results in large improvements. We propose a natural modification to N‐GMRES, which significantly improves the performance in a testing environment originally used to advocate N‐GMRES. Our proposed approach, which we refer to as O‐ACCEL (objective acceleration), is novel in that it minimizes an approximation to the objective function on subspaces of ${\mathbb{R}}^n$. We prove that O‐ACCEL reduces to the full orthogonalization method for linear systems when the objective is quadratic, which differentiates our proposed approach from existing acceleration methods. Comparisons with the limited‐memory Broyden–Fletcher–Goldfarb–Shanno and nonlinear conjugate gradient methods indicate the competitiveness of O‐ACCEL. As it can be combined with domain‐specific optimizers, it may also be beneficial in areas where limited‐memory Broyden–Fletcher–Goldfarb–Shanno and nonlinear conjugate gradient methods are not suitable.