The extragradient (EG), introduced by G. M. Korpelevich in 1976, is a well-known method to approximate solutions of saddle-point problems and their extensions such as variational inequalities and monotone inclusions. Over the years, numerous variants of EG have been proposed and studied in the literature. Recently, these methods have gained popularity due to new applications in machine learning and robust optimization. In this work, we survey the latest developments in the EG method and its variants for approximating solutions of nonlinear equations and inclusions, with a focus on the monotonicity and co-hypomonotonicity settings. We provide a unified convergence analysis for different classes of algorithms, with an emphasis on sublinear best-iterate and last-iterate convergence rates. We also discuss recent accelerated variants of EG based on both Halpern fixed-point iteration and Nesterov's accelerated techniques. Our approach uses simple arguments and basic mathematical tools to make the proofs as elementary as possible, while maintaining generality to cover a broad range of problems.

翻译：1976年由G. M. Korpelevich提出的外梯度法（EG）是一种求解鞍点问题及其推广形式（如变分不等式和单调包含）的著名近似方法。多年来，文献中已提出并研究了EG的诸多变体。近年来，随着在机器学习和鲁棒优化中的新应用，这些方法重新受到广泛关注。本工作综述了EG方法及其变体在近似求解非线性方程与包含问题方面的最新进展，重点聚焦于单调性与协-次单调性设定。我们为不同算法类别提供了统一的收敛性分析，重点强调次线性最优迭代步与最后迭代步的收敛速率。同时讨论了基于Halpern不动点迭代和Nesterov加速技术的EG加速变体。我们的方法采用简洁论证和基本数学工具，在保持证明过程尽量初等化的同时，保证足够的一般性以涵盖广泛的问题类型。