Algorithms for general monotone mixed variational inequalities (Q1276343)

The paper deals with the iterative methods for solving general monotone mixed variational inequalities: \( \langle Tu,g(v)-g(u)\rangle+\varphi(g(v))-\varphi(g(u))\geq 0\) for all \(g(v)\in H,\) where \(T, g:H\to H, \varphi:H\to R\cup \{+\infty\}\) is a proper, convex and lower semicontinuous function. The iterative method is based on the fact that the function \(u\in H\) is a solution of the above mixed variational inequality if and only if \(u\) satisfies the relation \(g(u)=J_\varphi[g(u)-\rho Tu]\), where \(J_\varphi = (I+\rho \partial \varphi)^{-1}\) is the resolvent and \(\rho>0\). The main algorithm has the iterative form \(u_0\in H\), \(g(u_{n+1})=g(u_n)+\rho Tu_n-\rho Tu_{n+1}-\gamma R(u_n)\), \(n=0,1,2,\dots\;\). The convergence of the algorithm to a solution of the general mixed variational inequality in the case of an invertible operator \(g:H\to H\) is verified. Another algorithm is based on the general Wiener-Hopf equation \(Tg^{-1}P_K z+\rho^{-1}Q_Kz=0\) which holds if \(\varphi\) is the indicator function of a closed convex set \(K\subset H\) and \(J_\varphi\equiv P_K\), the projection of \(H\) onto \(K\).