A note on spectral properties of some gradient methods

Starting from the work by Barzilai and Borwein, gradient methods have gained a great amount of attention, and efficient low-cost schemes are available nowadays. The acceleration strategies used by these methods are based on the definition of effective steplength updating rules, which capture spectral properties of the Hessian of the objective function. The methods arising from this idea represent effective computational tools, extremely appealing for a variety of large-scale optimization problems arising in applications. In this work we discuss the spectral properties of some recently proposed gradient methods with the aim of providing insight into their computational effectiveness. Numerical experiments supporting and illustrating the theoretical analysis are provided.