Well-Posedness of Inverse Variational Inequalities

Let $\Omega\subset R^P$ be a nonempty closed and convex set and $f:R^P\to R^P$ be a function. The inverse variational inequality is to find $x^*\in R^P$ such that $$f(x^*)\in \Omega,\quad \langle f'-f(x^*),x^*\rangle\ge 0,\quad \forall f'\in \Omega.$$ The purpose of this paper is to investigate the well-posedness of the inverse variational inequality. We establish some characterizations of its well-posedness. We prove that under suitable conditions, the well-posedness of an inverse variational inequality is equivalent to the existence and uniqueness of its solution. Finally, we show that the well-posedness of an inverse variational inequality is equivalent to the well-posedness of an enlarged classical variational inequality.