For more (non exhaustive) informations regarding the proximity operator and the associated proximal algorithms:
F. Bach, R. Jenatton, J. Mairal, G. Obozinski:
Optimization with sparsity-inducing penalties.
Found. Trends in Machine Learn., vol. 4, no. 1, pp. 1–106, 2012.
[pdf]
H. H. Bauschke and P. L. Combettes:
Convex Analysis and Monotone Operator Theory in Hilbert Spaces (2nd Edition).
Springer, New York, 2017.
H. H. Bauschke, J. M. Borwein, P. L. Combettes:
Bregman monotone optimization algorithms.
SIAM Journal on Control and Optimization, vol. 42, no. 2, pp. 596-636, Jun. 2003. [pdf]
L. M. Briceño-Arias and P. L. Combettes:
Convex variational formulation with smooth coupling for multicomponent signal decomposition and recovery.
Numerical Mathematics: Theory, Methods, and Applications, vol. 2, no. 4, pp. 485-508, November 2009. [pdf]
P. L. Combettes:
Convexité et signal.
Actes du Congrès de Mathématiques Appliquées et Industrielles, pp. 6-16, Pompadour, France, May 28-June 1, 2001.[pdf]
P. L. Combettes and V. R. Wajs:
Signal Recovery by Proximal Forward-Backward Splitting.
Multiscale Modeling and Simulation, vol. 4, no. 4, pp. 1168–1200, 2005.[pdf]
P. L. Combettes and J.-C. Pesquet:
A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery.
IEEE Journal of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, December 2007. [pdf]
P. L. Combettes and J.-C. Pesquet:
Proximal Splitting Methods in Signal Processing.
In Fixed-Point Algorithms for Inverse Problems in Science and Engineering, H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, Eds., pp. 185–212. Springer-Verlag, New York, 2010.
[pdf]
P. L. Combettes, L. Condat, J.-C. Pesquet and B. C. Vu:
A forward-backward view of some primal-dual optimization methods in image recovery.
In Proceedings of IEEE International Conference on Image Processing (ICIP 2014), Paris, France, 27-30 October 2014.
[pdf]
J. B. Hiriart-Urruty and C. Lemaréchal:
Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods.
Vol. 306 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York, 1993.
N. Komodakis and J.-C. Pesquet:
Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization problems.
IEEE Signal Processing Magazine, vol. 32, no. 6, pp. 31–54, Nov. 2014. [pdf]
J. J. Moreau:
Fonctions convexes duales et points proximaux dans un espace hilbertien.
Comptes Rendus de l'Académie des Sciences de Paris, vol. A255, no. 22, pp. 2897-2899, Nov. 1962.
N. Parikh and S. Boyd:
Proximal Algorithms.
Foundations and Trends in Optimization, vol. 1, no. 3, Jan. 2014. [pdf]
G. Peyré:
The Numerical Tours of Signal Processing - Advanced Computational Signal and Image Processing.
IEEE Computing in Science and Engineering, vol. 13, no. 4, pp. 94–97, 2011.