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We study the variable metric forward-backward splitting algorithm for convexminimization problems without the standard assumption of the Lipschitzcontinuity of the gradient. In this setting, we prove that, by requiring onlymild assumptions on the smooth part of the objective function and using severaltypes of line search procedures for determining either the gradient descentstepsizes, or the relaxation parameters, one still obtains weak convergence ofthe iterates and convergence in the objective function values. Moreover, the$o(1/k)$ convergence rate in the function values is obtained if slightlystronger differentiability assumptions are added. We also illustrate severalapplications including problems that involve Banach spaces and functions ofdivergence type.

The variable metric forward-backward splitting algorithm under mild differentiability assumptions