Let $Q \colon \mathbb{R}^n \to \mathbb{R}$ denote a convex function with $g(x) = \nabla Q(x)$ well-defined. I am interested in defining the following variables \begin{eqnarray} c & = & x - g(x) \\ d & = & x + g(x) \end{eqnarray} and then writing $c$ as a function of $d$. My approach is as follows:
- Let $h(x) = x + g(x)$. We need to show that the function $h^{-1}$ is well-defined.
- Apply chain rule. Using the notation $[J_{x}c](d)$ to denote the Jacobian of $c$ with respect to $x$ evaluated at $d$, we have \begin{eqnarray} [J_d c](d) & = & [J_x c](h^{-1}(d)) \ [J_d h^{-1}](d). \end{eqnarray}
- Standard methods of finding a primitive function that is consistent with the Jacobian $[J_d c](d)$ may be applied.
My main concern is with step 1. My attempt is as follows. We assumed $Q$ to be convex, so each entry $g_i(x)$ is non-decreasing in $x_j$ for each $j$. The entry $h_i$ is the sum of $x_i$ which is strictly increasing and $g_i$ which is non-decreasing. Therefore $h^{-1}$ is well defined and $x = h^{-1}(d)$. Is this reasoning correct?
"so each entry $g_i(x)$ is non-decreasing in $x_j$ for each $j$": This is not true.
But the function $g$ is monotone and, thus, $h$ is strongly monotone. Invertibility of $h$ follows then from the theorem of Browder-Minty on monotone operators.