  In this paper, by using tools of second-order variational analysis, we study
the popular forward-backward splitting method with Beck-Teboulle's line-search
for solving convex optimization problem where the objective function can be
split into the sum of a differentiable function and a possible nonsmooth
function. We first establish that this method exhibits global convergence to an
optimal solution of the problem (if it exists) without the usual assumption
that the gradient of the differentiable function involved is globally Lipschitz
continuous. We also obtain the $o(k^{-1})$ complexity for the functional value
sequence when this usual assumption is weaken from global Lipschitz continuity
to local Lipschitz continuity, improving the existing $\mathcal{O}(k^{-1})$
complexity result. We then derive the local and global Q-linear convergence of
the method in terms of both the function value sequence and the iterative
sequence, under a general metric subregularity assumption which is
automatically satisfied for convex piecewise-linear-quadratic optimization
problems. In particular, we provide verifiable sufficient conditions for metric
subregularity assumptions, and so, local and global Q-linear convergence of the
proposed method for broad structured optimization problems arise in machine
learning and signal processing including Poisson linear inverse problem, the
partly smooth optimization problems, as well as the $\ell_1$-regularized
optimization problems. Our results complement the current literature by
providing $Q$-linear convergence result to the forward-backward splitting
method under weaker assumptions. Moreover, via this approach, we obtain several
full characterizations for the uniqueness of optimal solution to Lasso problem,
which covers some recent results in this direction.
