  The paper proposes a linesearch for a primal-dual method. Each iteration of
the linesearch requires to update only the dual (or primal) variable. For many
problems, in particular for regularized least squares, the linesearch does not
require any additional matrix-vector multiplications. We prove convergence of
the proposed method under standard assumptions. We also show an ergodic
$O(1/N)$ rate of convergence for our method. In case one or both of the
prox-functions are strongly convex, we modify our basic method to get a better
convergence rate. Finally, we propose a linesearch for a saddle point problem
with an additional smooth term. Several numerical experiments confirm the
efficiency of our proposed methods.
