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.

翻译：我们研究使用逆向调和雷利利商数,目标是逐步选择非线性、不受限制的优化问题梯度方法,这不仅提供了一个优雅和灵活的框架,以调整和重新解释现有的逐步化计划,而且还激励了新的灵活和可金枪鱼的等长家庭,特别是我们分析并推广适应性的巴齐莱-波文方法,将其推广到一个新的阶梯式家庭。这个家庭利用负面价值来对付目标,我们也考虑积极的目标。我们提出对大和廖扩大结果的二次问题的趋同分析(2002年),并进行有关方法潜力的实验。