  We study the variable metric forward-backward splitting algorithm for convex
minimization problems without the standard assumption of the Lipschitz
continuity of the gradient. In this setting, we prove that, by requiring only
mild assumptions on the smooth part of the objective function and using several
types of line search procedures for determining either the gradient descent
stepsizes, or the relaxation parameters, one still obtains weak convergence of
the iterates and convergence in the objective function values. Moreover, the
$o(1/k)$ convergence rate in the function values is obtained if slightly
stronger differentiability assumptions are added. We also illustrate several
applications including problems that involve Banach spaces and functions of
divergence type.
