For problems of unconstrained optimization, the concept of inexact oracle proposed by O. Devolder, F. Glineur and Yu.E. Nesterov is well known. We introduce an analog of the concept of inexact oracle (model of a function) for abstract equilibrium problems, variational inequalities, and saddle-point problems. This allows us to propose an analog of Nemirovskii’s known mirror prox method for variational inequalities with an adaptive adjustment to the smoothness level for a fairly wide class of problems. The auxiliary problems at the iterations of the method can be solved with error. It is shown that the resulting errors do not accumulate during the operation of the method. Estimates of the convergence rate of the method are obtained, and its optimality from the viewpoint of the theory of lower oracle estimates is established. It is shown that the method is applicable to mixed variational inequalities and composite saddle-point problems. An example showing the possibility of an essential acceleration of the method as compared to the theoretical estimates due to the adaptivity of the stopping rule is given.

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关于一类变分不等式的自适应近邻方法及相关问题

对于无约束优化的问题，由O. Devolder，F。Glineur和Yu.E.提出了不精确预言的概念。内斯特罗夫是众所周知的。对于抽象的平衡问题，变分不等式和鞍点问题，我们引入了不精确的oracle（函数模型）概念的类似物。这使我们能够提出Nemirovskii已知的镜像代理方法的类似方法来解决变分不等式，并针对相当广泛的问题对平滑度水平进行自适应调整。该方法迭代中的辅助问题可以通过错误解决。结果表明，所产生的误差在该方法的运行期间不会累积。获得了该方法的收敛速度的估计，并从较低的预言估计的理论角度确定了该方法的最优性。结果表明，该方法适用于混合变分不等式和复合鞍点问题。给出了一个示例，该示例显示了由于停止规则的适应性而使方法与理论估计值相比有必要加速的可能性。