# Generalized Conditional Gradient with Augmented Lagrangian for Composite Minimization

## TL;DR

This paper introduces CGALP, a hybrid optimization algorithm combining generalized conditional gradient and augmented Lagrangian methods, capable of efficiently solving complex convex composite problems with affine constraints.

## Contribution

The paper proposes a novel splitting scheme, CGALP, that relaxes Lipschitz gradient requirements and handles multiple convex functions with affine constraints using an augmented Lagrangian approach.

## Key findings

- Proves asymptotic feasibility and boundedness of dual multipliers.
- Establishes convergence of Lagrangian values to saddle-point optimality.
- Provides convergence rates for feasibility gap and Lagrangian values.

## Abstract

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.