Some numerical algorithms for monotone variational inequalities

Abstract

This dissertation presents some numerical algorithms for monotone variational inequalities (VI) in the finite-dimensional space Rm. First, some projection-type algorithms are presented. The applications to the complementarity problems, the p-norm Steiner Minimum Trees problems and the tra±c equilibrium prob- lems show that these projection-type methods are effective in practice. These techniques for VI are then extended to finding a root of a maximal monotone op- erator; therefore a hybrid extragradient-proximal algorithm is presented. Then, the well-known proximal point algorithm is applied to solve the sub-VIs of the augmented Lagrangian method for monotone VI with linear constraints; and thus a proximal augmented Lagrangian method is presented. An inexact version of the proximal augmented Lagrangian method solving the sub-VIs approximately is also provided. Finally, some inexact proximal-type decomposition algorithms are presented for monotone VI with separate structures. In particular, some new practical inexact criteria for solving the sub-VIs are presented. Finally, some conclusions are made and some directions for future research are presented.