  The main result of the present theoretical paper is an original decomposition
formula for the proximal operator of the sum of two proper, lower
semicontinuous and convex functions $f$ and $g$. For this purpose, we introduce
a new operator, called $f$-proximal operator of $g$ and denoted by
$\mathrm{prox}^f_g$, that generalizes the classical notion. Then we prove the
decomposition formula $\mathrm{prox}_{f+g} = \mathrm{prox}_f \circ
\mathrm{prox}^f_g$. After collecting several properties and characterizations
of $\mathrm{prox}^f_g$, we prove that it coincides with the fixed points of a
generalized version of the classical Douglas-Rachford operator. This
relationship is used for the construction of a weakly convergent algorithm that
computes numerically this new operator $\mathrm{prox}^f_g$, and thus, from the
decomposition formula, allows to compute numerically $\mathrm{prox}_{f+g}$. It
turns out that this algorithm was already considered and implemented in
previous works, showing that $\mathrm{prox}^f_g$ is already present (in a
hidden form) and useful for numerical purposes in the existing literature.
However, to the best of our knowledge, it has never been explicitly expressed
in a closed formula and neither been deeply studied from a theoretical point of
view. The present paper contributes to fill this gap in the literature. Finally
we give an illustration of the usefulness of the decomposition formula in the
context of sensitivity analysis of linear variational inequalities of second
kind in a Hilbert space.
