An Algorithm Based on Resolvant Operators for Solving Positively Semidefinite Variational Inequalities
Department of Applied Mathematics, Dalian University of Technology, Dalian, Liaoning 116024, China
Fixed Point Theory and Applications 2007, 2007:076040 doi:10.1155/2007/76040Published: 4 November 2007
A new monotonicity, -monotonicity, is introduced, and the resolvant operator of an -monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality (VI) problem VI is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method is given for calculating -solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.