  In this paper we propose a splitting scheme which hybridizes generalized
conditional gradient with a proximal step which we call CGALP algorithm, for
minimizing the sum of three proper convex and lower-semicontinuous functions in
real Hilbert spaces. The minimization is subject to an affine constraint, that
allows in particular to deal with composite problems (sum of more than three
functions) in a separate way by the usual product space technique. While
classical conditional gradient methods require Lipschitz-continuity of the
gradient of the differentiable part of the objective, CGALP needs only
differentiability (on an appropriate subset), hence circumventing the intricate
question of Lipschitz continuity of gradients. For the two remaining functions
in the objective, we do not require any additional regularity assumption. The
second function, possibly nonsmooth, is assumed simple, i.e., the associated
proximal mapping is easily computable. For the third function, again nonsmooth,
we just assume that its domain is also bounded and that a linearly perturbed
minimization oracle is accessible. In particular, this last function can be
chosen to be the indicator of a nonempty bounded closed convex set, in order to
deal with additional constraints. Finally, the affine constraint is addressed
by the augmented Lagrangian approach. Our analysis is carried out for a wide
choice of algorithm parameters satisfying so called "open loop" rules. As main
results, under mild conditions, we show asymptotic feasibility with respect to
the affine constraint, boundedness of the dual multipliers, and convergence of
the Lagrangian values to the saddle-point optimal value. We also provide
(subsequential) rates of convergence for both the feasibility gap and the
Lagrangian values.
