Interior point methods have been known for decades to be useful for the resolution of small to medium size constrained optimization problems. These approaches have the benefit of ensuring feasibility of the iterates through a logarithmic barrier. We propose to incorporate a proximal forward-backward step in the resolution of the barrier subproblem to account for non-necessarily differentiable terms arising in the objective function. The combination of this scheme with a novel linesearch strategy gives rise to the so-called Proximal Interior Point Algorithm (PIPA) suitable for the minimization of the sum of a smooth convex function and a non-smooth convex one under general convex constraints. The convergence of PIPA is secured under mild assumptions.
As demonstrated by numerical experiments carried out on a large-scale hyperspectral image unmixing application, the proposed method outperforms the state-of-the-art.