  Based on differences of points and differences of gradients over the most
recent three iterations, together with the Taylor's theorem, two forms of the
quasi-Newton equations at the recent iteration are constructed. By using the
two forms of the quasi-Newton equation and the method of least squares,
three-point step size gradient methods for solving unconstrained optimization
problem are proposed. It is proved by using the relaxed generalized Armijo step
size rule that the new method is of global convergence properties if the
gradient function is uniformly continuous. Moreover, it is shown that, when the
objective function is pseudo-convex (quasi-convex) function, the new method has
strong convergence results. In addition, it is also shown under some suitable
assumptions that the new method is of super-linear and linear convergence.
Although multi-piont information is used, TBB has the feature of simplicity,
low memory requirement and only first order information being used, the new
method is very suitable for solving large-scale optimization problems.
Numerical experiments are provided and the efficiency, robustness and analysis
of TBB are confirmed.
