A penalization-gradient algorithm for variational inequalities (Q554800)

Summary: This paper is concerned with the study of a penalization-gradient algorithm for solving variational inequalities, namely, find \(\bar x \in C\) such that \(\langle A\bar x, y - \bar x \rangle \geq 0\) for all \(y \in C\), where \(A : H \rightarrow H\) is a single-valued operator, \(C\) is a closed convex set of a real Hilbert space \(H\). Given \(\Psi : H \rightarrow \mathbb R \cup \{ +\infty \}\) which acts as a penalization function with respect to the constraint \(\bar x \in C\), and a penalization parameter \(\beta_k\), we consider an algorithm which alternates a proximal step with respect to \(\partial\Psi\) and a gradient step with respect to \(A\) and reads as \(x_k = (I + \lambda_k\beta_k\partial\Psi)^{-1}(x_{k-1} - \lambda_kAx_{k-1})\). Under mild hypotheses, we obtain weak convergence for an inverse strongly monotone operator and strong convergence for a Lipschitz continuous and strongly monotone operator. Applications to hierarchical minimization and fixed-point problems are also given and the multivalued case is reached by replacing the multivalued operator by its Yosida approximate which is always Lipschitz continuous.