Proximal Algorithm Meets a Conjugate descent

This paper proposes an enhancement of the non linear conjugate gradient algorithm for some non-smooth problems. We first extend some results of descent algorithms in the smooth case for convex non-smooth functions. We then construct a conjugate descent algorithm based on the proximity operator to obtain a descent direction. We finally provide a convergence analysis of this algorithm, even when the proximity operator must be computed by an iterative process. Numerical experiments show that this kind of method has some potential, even if proposed algorithms do not outperform accelerated first order algorithm yet.