Self-adaptive projection-based prediction-correction method for constrained variational inequalities

The problems studied in this paper are a class of monotone constrained variational inequalities VI (S, f) in which S is a convex set with some linear constraints. By introducing Lagrangian multipliers to the linear constraints, such problems can be solved by some projection type prediction-correction methods. We focus on the mapping f that does not have an explicit form. Therefore, only its function values can be employed in the numerical methods. The number of iterations is significantly dependent on a parameter that balances the primal and dual variables. To overcome potential difficulties, we present a self-adaptive prediction-correction method that adjusts the scalar parameter automatically. Convergence of the proposed method is proved under mild conditions. Preliminary numerical experiments including some traffic equilibrium problems indicate the effectiveness of the proposed methods.