On the adaptive proximal method for a class of variational inequalities and related problems

For problems of unconstrained optimization, the concept of inexact oracle proposed by O. Devolder, F. Gleener and Yu. E. Nesterov is well known. We introduce an analog of the notion 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 proximal method for variational inequalities with an adaptive adjustment to the level of smoothness for a fairly wide class of problems. It is also possible to inexactly solve auxiliary problems at the iterations of the method. 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 bounds 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.