New Active-Set Frank-Wolfe Variants for Minimization over the Simplex and the L1- Ball

In this paper, we are concerned with minimization problems over the unit simplex. Here, we propose the use of an active-set estimate that enables us to define an algorithmic framework where the variables estimated active and those estimated non-active are updated separately at each iteration. In particular, we consider different variants of the Frank-Wolfe direction to be combined with the proposed active-set strategy, proving the convergence of the algorithm for each of them. Then, we focus on the problem of minimizing a function over the `1ball, showing how our algorithmic framework can be efficiently adapted to this problem. Preliminary numerical results show the benefits of the proposed active-set estimate.