Generalized conditional gradient with augmented lagrangian for composite minimization

In this paper we propose a splitting scheme which hybridizes the generalized conditional gradient with a proximal step and which we call the 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, in particular, allows one to deal with composite problems (a sum of more than three functions) in a separable 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) and hence circumvents 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 weakly compact 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, weak convergence of the dual multipliers, and convergence of the Lagrangian values to the saddle-point optimal value. We also provide pointwise and ergodic rates of convergence for both the feasibility gap and the Lagrangian values.