# What is an extragradient method?

I've searched Google, but it seems that only research journal papers appear in search results, where some new, improved, or specialized extragradient method is discussed. I've also searched Wikipedia and Wolfram MathWorld.

I would like to perhaps know the straightforward definition of that, instead of deducing it from a number of articles, where the basic/original method is not defined.

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Briefly, extragradient methods include an extrapolation step for the evaluation of the gradient for the next iteration, e.g., \begin{aligned} \bar x^{k} &= x^k + \tau(x^k - x^{k-1}),\\ x^{k+1} &= x^k + \gamma_k \nabla f(\bar x^k), \end{aligned} where $\gamma_k>0$ is a suitable step size and $\tau\in (0,1)$ is an extrapolation parameter.