  In this paper, we propose AdaBB, an adaptive gradient method based on the
Barzilai-Borwein stepsize. The algorithm is line-search-free and
parameter-free, and essentially provides a convergent variant of the
Barzilai-Borwein method for general unconstrained convex optimization. We
analyze the ergodic convergence of the objective function value and the
convergence of the iterates for solving general unconstrained convex
optimization. Compared with existing works along this line of research, our
algorithm gives the best lower bounds on the stepsize and the average of the
stepsizes. Moreover, we present an extension of the proposed algorithm for
solving composite optimization where the objective function is the summation of
a smooth function and a nonsmooth function. Our numerical results also
demonstrate very promising potential of the proposed algorithms on some
representative examples.
