  We study the use of inverse harmonic Rayleigh quotients with target for the
stepsize selection in gradient methods for nonlinear unconstrained optimization
problems. This provides not only an elegant and flexible framework to
parametrize and reinterpret existing stepsize schemes, but also gives
inspiration for new flexible and tunable families of steplengths. In
particular, we analyze and extend the adaptive Barzilai-Borwein method to a new
family of stepsizes. While this family exploits negative values for the target,
we also consider positive targets. We present a convergence analysis for
quadratic problems extending results by Dai and Liao (2002), and carry out
experiments outlining the potential of the approaches.
