Regularized Estimating Equations from the Fixed-point and Variational Inequality Perspectives

In this working note, we make some observations about the equivalences between regularized estimating equations, fixed-point problems and variational inequalities. A summary of our findings is given below: (a) A regularized estimating equation is equivalent to a fixed-point problem, specified by the proximal operator of the corresponding penalty; (b) A regularized estimating equation is equivalent to a generalized variational inequality; (c) Both equivalences extend to any estimating equations and any penalty functions. To our knowledge, these observations have never been presented in the literature before. We hope our new findings can lead to further research in both computational and theoretical aspects of regularized estimating equations.