We make some observations about the equivalences between regularized estimating equations, fixed-point problems and variational inequalities: (a) A regularized estimating equation is equivalent to a fixed-point problem, specified via the proximal operator of the corresponding penalty. (b) A regularized estimating equation is equivalent to a (generalized) variational inequality. Both equivalences extend to any estimating equations with convex penalty functions. To solve large-scale regularized estimating equations, it is worth pursuing computation by exploiting these connections. While fast computational algorithms are less developed for regularized estimating equation, there are many efficient solvers for fixed-point problems and variational inequalities. In this regard, we apply some efficient and scalable solvers which can deliver hundred-fold speed improvement. These connections can lead to further research in both computational and theoretical aspects of the regularized estimating equations.