  We consider the extragradient method to minimize the sum of two functions,
the first one being smooth and the second being convex. Under the
Kurdyka-Lojasiewicz assumption, we prove that the sequence produced by the
extragradient method converges to a critical point of the problem and has
finite length. The analysis is extended to the case when both functions are
convex. We provide, in this case, a sublinear convergence rate, as for
gradient-based methods. Furthermore, we show that the recent small-prox
complexity result can be applied to this method. Considering the extragradient
method is an occasion to describe an exact line search scheme for proximal
decomposition methods. We provide details for the implementation of this scheme
for the one norm regularized least squares problem and demonstrate numerical
results which suggest that combining nonaccelerated methods with exact line
search can be a competitive choice.
