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.

本文研究了解决两个函数之和的最小值问题的外推梯度方法，证明了在 Kurdyka-Lojasiewicz 假设下，该方法得到的序列收敛于问题的临界点并具有有限长度。该分析拓展到两个函数均为凸函数的情况，并证明了其次线性收敛率。此外，我们展示了将小型 prox 复杂度结果应用于此方法的方法。这个方法提供了一个机会来描述一种精确的线搜索方案，用于 proximal 分解方法。我们提供了实现这种方案对于一范数规则化最小二乘问题的细节，并展示了数值结果，这些结果表明将非加速方法与精确线搜索相结合可能是一个有竞争力的选择。