A class of projection-contraction methods applied to monotone variational inequalities (Q5931722)

Approximation methods for the variational inequality NEWLINE\[NEWLINE\langle T(x),v-g(x)\rangle \geq 0,\qquad v\in J(x)NEWLINE\]NEWLINE in a Hilbert space are studied where \(J\) attains closed convex values. It is assumed that the canonical projections \(P_{J(x)}\) onto the set \(J(x)\) are known. In this case, the solutions of the variational inequality are those points which satisfy \(g(x)=P_{J(x)}[g(x)-tT(x)]\) for some \(t>0\), and iterative algorithms can be used to approximate a solution. Convergence of these algorithms are proved under Lipschitz and monotonicity assumptions on \(T\) and under the assumption that \(g\) is Lipschitz and expanding.