Projection-type Algorithms For Variational Inequalities

Variational inequality is a kind of very important nonlinear problem, which has been widely applied to economics, mechanics, applied science and other fields. Network resource allocation, image recovery and other practical problems can be expressed as variational inequalities problems. This thesis mainly discusses two kinds of subgradient methods and an extragradient method for finding common elements of the set of solutions of variational inequalities and the set of fixed points of operators in Hilbert space on the basis of the existing projection algorithm. We do the specific research in the following aspects.(1)we suggest two new iterative methods for finding a common element of the set of solutions of variational inequality problem and the set of fixed points of a contraction mapping. We also present the weak convergence theorem and strong convergence theorem for these new methods, privided that the fixed point mapping is a strict pseudo-contraction and the mapping associated with the variational inequality problem is monotone. On the other hand, the second iterate is a projection onto a half space whose bounding hyperplane supports the feasible set at a certain point in our methods. Moreover, the results presented in this chapter improve and unify important recent results announced by many authors.(2)we present a hybrid extragradient method for solving variational inequality problem. Under the assumption that the mapping associated with the variational inequality problem is continuous and monotone, we prove that the sequence generated by our method converges strongly to the common element of solutions of variational inequality problem and fixed point set of a countably infinite family of nonexpansive mappings.