Splitting methods for pseudomonotone mixed variational inequalities
The author analyzes some algorithms for solving a monotone mixed variational inequality \(\;\langle Tu,v-u\rangle + \varphi (v)-\varphi(u)\geq 0,\quad\) for all\(\;v\in H\) with a Hilbert space \(H\), a nonlinear operator \(T:H\to H\) and a lower semicontinuous function \(\varphi:H\to R\cup \{+\infty\}.\) The resolvent operator technique is used to study a number of splitting methods for solving the mixed variational inequalities. The methods are based on the fact that a solution \(u\in H\) of a variational inequality satisfies the relation \(u=J_\varphi [u-\rho Tu] \), where \(J_\varphi= (I+\rho \partial \varphi)^{-1}\) is the resolvent operator. The resolvent operator method requires the assumption that \(T\) must be strongly monotone for the convergence. In order to overcome this difficulty the author suggests the double resolvent formula \(\;u=J_\varphi \Big\lgroup u-\rho TJ_\varphi [u-\rho Tu]\Big\rgroup.\) Several splitting methods for solving the iteration method connected with fixed point algorithms are analyzed.
scientific article; zbMATH DE number 1496350
scientific article