Extragradient method (EG) Korpelevich [1976] is one of the most popular methods for solving saddle point and variational inequalities problems (VIP). Despite its long history and significant attention in the optimization community, there remain important open questions about convergence of EG. In this paper, we resolve one of such questions and derive the first last-iterate $O(1/K)$ convergence rate for EG for monotone and Lipschitz VIP without any additional assumptions on the operator. The rate is given in terms of reducing the squared norm of the operator. Moreover, we establish several results on the (non-)cocoercivity of the update operators of EG, Optimistic Gradient Method, and Hamiltonian Gradient Method, when the original operator is monotone and Lipschitz.
翻译:Korpelevich[1976年]是解决马鞍点和变异不平等问题最受欢迎的方法之一。尽管其历史悠久,而且优化社区也十分关注,但仍存在着关于环境趋同的重要未决问题。在本文件中,我们解决了其中一个问题,并得出了单体酮和利普西茨贵宾的EG最接近率,而没有就操作者作出任何额外的假设。这一比率是以降低操作者的正方规范为单位的。此外,我们还就EG、偏重重力法和汉密尔顿重力法(原操作者为单体和利普西茨)的最新操作者(非)协调性得出了几项结果。