Résumé : This paper presents new algorithms to minimize total variation and more generally $l^1$-norms under a general convex constraint. The algorithms are based on a recent advance in convex optimization proposed by Yurii Nesterov. Depending on the regularity of the data fidelity term, we solve either a primal problem, either a dual problem. First we show that standard first order schemes allow to get solutions of precision $\epsilon$ in $O(\frac{1}{\epsilon^2})$ iterations at worst. For a general convex constraint, we propose a scheme that allows to obtain a solution of precision $\epsilon$ in $O(\frac{1}{\epsilon})$ iterations. For a strongly convex constraint, we solve a dual problem with a scheme that requires $O(\frac{1}{\sqrt{\epsilon}})$ iterations to get a solution of precision $\epsilon$. Thus, depending on the regularity of the data term, we gain from one to two orders of magnitude in the convergence rates with respect to standard schemes. Finally we perform some numerical experiments which confirm the theoretical results on various problems.