  The Barzilai-Borwein (BB) method is a popular and efficient tool for solving
large-scale unconstrained optimization problems. Its search direction is the
same as for the steepest descent (Cauchy) method, but its stepsize rule is
different. Owing to this, it converges much faster than the Cauchy method. A
feature of the BB method is that it may generate too long steps, which throw
the iterates too far away from the solution. Moreover, it may not converge,
even when the objective function is strongly convex. In this paper, a
stabilization technique is introduced. It consists in bounding the distance
between each pair of successive iterates, which often allows for decreasing the
number of BB iterations. When the BB method does not converge, our simple
modification of this method makes it convergent. Under suitable assumptions, we
prove its global convergence, despite the fact that no line search is involved,
and only gradient values are used. Since the number of stabilization steps is
proved to be finite, the stabilized version inherits the fast local convergence
of the BB method. The presented results of extensive numerical experiments show
that our stabilization technique often allows the BB method to solve problems
in a fewer iterations, or even to solve problems where the latter fails.
