Splitting method with adaptive step-size

Abstract

© Springer Nature Switzerland AG 2019. We suggest the modified splitting method for mixed variational inequalities and prove its convergence under rather mild assumptions. This method maintains the basic convergence properties but does not require any iterative step-size search procedure. It involves a simple adaptive step-size choice, which takes into account the problem behavior along the iterative sequence. The key element of this approach is a given majorant step-size sequence converging to zero. The next decreased value of step-size is taken only when the current iterate does not give a sufficient descent of the objective function. This descent value is estimated with the help of an Armijo-type condition, similar to the rule used in the inexact step-size linesearch. If the current iterate gives a sufficient descent, we can even take an increasing step-size value at the next iterate. Preliminary results of computational experiments confirm the efficiency of the proposed modification in comparison with the ordinary splitting method using the inexact step-size linesearch procedure.