Journal of Convex Analysis 26 (2019), No. 3, 699--718
Copyright Heldermann Verlag 2019
On a Decomposition Formula for the Proximal Operator of the Sum of Two Convex Functions
Samir Adly
Institut de Recherche XLIM, Université de Limoges, 123 avenue A. Thomas, 87060 Limoges, France
samir.adly@unilim.fr
Loic Bourdin
Institut de Recherche XLIM, Université de Limoges, 123 avenue A. Thomas, 87060 Limoges, France
loic.bourdin@unilim.fr
Fabien Caubet
Institut de Mathématiques, Université de Toulouse, 118 route de Narbonne, 31062 Toulouse, France
fabien.caubet@math.univ-toulouse.fr
[Abstract-pdf]
\def\prox{\mathrm{prox}} 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 {\it $f$-proximal operator of~$g$} and denoted by $\prox^f_g$, that generalizes the classical notion. Then we prove the decomposition formula~$\prox_{f+g} = \prox_f \circ \prox^f_g$. After collecting several properties and characterizations of~$\prox^f_g$, 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 $\prox^f_g$, and thus, from the decomposition formula, allows to compute numerically $\prox_{f+g}$. It turns out that this algorithm was already considered and implemented in previous works, showing that~$\prox^f_g$ 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.
Keywords: Convex analysis, proximal operator, Douglas-Rachford operator, forward-backward operator.
MSC: 46N10, 47N10, 49J40, 49Q12
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