On a decomposition formula for the proximal operator of the sum of two convex functions

TL;DR

This paper introduces a new decomposition formula for the proximal operator of the sum of two convex functions, generalizing classical notions and providing a theoretical foundation for numerical algorithms.

Contribution

It defines a new operator, the $f$-proximal operator of $g$, and proves a decomposition formula, enriching the theoretical understanding and computational methods for convex optimization.

Findings

Introduces the $f$-proximal operator of $g$ as a generalization.

Proves the decomposition formula $ ext{prox}_{f+g} = ext{prox}_f ext{circ} ext{prox}_g^f$.

Provides a weakly convergent algorithm for computing the new operator.

Abstract

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.