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 ff and gg. For this purpose, we introduce a new operator, called ff-proximal operator of gg and denoted by proxgf\prox^f_g, that generalizes the classical notion. Then we prove the decomposition formula proxf+g=proxf∘proxgf\prox_{f+g} = \prox_f \circ \prox^f_g. After collecting several properties and characterizations of proxgf\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 proxgf\prox^f_g, and thus, from the decomposition formula, allows to compute numerically proxf+g\prox_{f+g}. It turns out that this algorithm was already considered and implemented in previous works, showing that proxgf\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.

Author information

Contact details are reproduced from the original publication and may be historical.

Samir Adly

Institut de Recherche XLIM, Université de Limoges, 123 avenue A. Thomas, 87060 Limoges, France

samir.adly@unilim.fr

Loïc Bourdin

Institut de Recherche XLIM, Université de Limoges, 123 avenue A. Thomas, 87060 Limoges, France

loic.bourdin@unilim.fr

Keywords

  • Convex analysis
  • proximal operator
  • Douglas-Rachford operator
  • forward-backward operator

Mathematics Subject Classification

46N10, 47N10, 49J40, 49Q12

References

  1. H. Attouch, H. Brezis: Duality for the sum of convex functions in general Banach spaces, in: Aspects of Mathematics and its Applications, North-Holland Math. Library 34, North-Holland, Amsterdam (1986) 125–133.https://doi.org/10.1016/S0924-6509(09)70252-1Google Scholar
  2. H. H. Bauschke, P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edition, CMS Books in Mathematics, Springer, New York (2017).https://doi.org/10.1007/978-3-319-48311-5Google Scholar
  3. A. Bensoussan, J.-L. Lions: Inéquations quasi-variationnelles dépendant d’un paramètre, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4(2) (1977) 231–255.Google Scholar
  4. P. L. Combettes, D. Dũng, B. C. Vũ: Dualization of signal recovery problems, Set-Valued Var. Anal. 18(3-4) (2010) 373–404.https://doi.org/10.1007/s11228-010-0147-7Google Scholar
  5. P. L. Combettes, J.-C. Pesquet: Proximal splitting methods in signal processing, in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optim. Appl. 49, Springer, New York (2011) 185–212.https://doi.org/10.1007/978-1-4419-9569-8_10Google Scholar
  6. J. Douglas, H. H. Rachford: On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc. 82 (1956) 421–439.https://doi.org/10.1090/S0002-9947-1956-0084194-4Google Scholar
  7. I. Ekeland, R. Temam: Analyse Convexe et Problèmes Variationnels, Collection Études Mathématiques, Dunod-Gauthier-Villars, Paris et al. (1974).Google Scholar
  8. A. Haraux: How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan 29(4) (1977) 615–631.https://doi.org/10.2969/jmsj/02940615Google Scholar
  9. J.-B. Hiriart-Urruty, C. Lemaréchal: Fundamentals of Convex Analysis, Grundlehren Text Editions, Springer, Berlin (2001).https://doi.org/10.1007/978-3-642-56468-0Google Scholar
  10. F. Mignot: Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis 22(2) (1976) 130–185.https://doi.org/10.1016/0022-1236(76)90017-3Google Scholar
  11. J.-J. Moreau: Fonctions convexes duales et points proximaux dans un espace hilbertien, C. R. Acad. Sci. Paris 255 (1962) 2897–2899.Google Scholar
  12. J.-J. Moreau: Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965) 273–299.https://doi.org/10.24033/bsmf.1625Google Scholar
  13. Y. Qiu, T. L. Magnanti: Sensitivity analysis for variational inequalities, Math. Oper. Res. 17(1) (1992) 61–76.https://doi.org/10.1287/moor.17.1.61Google Scholar
  14. R. T. Rockafellar: Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton (1970).Google Scholar
  15. R. T. Rockafellar: Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2(3) (1985) 167–184.https://doi.org/10.1016/s0294-1449(16)30401-2Google Scholar
  16. R. T. Rockafellar: Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 449–482.https://doi.org/10.1016/s0294-1449(17)30034-3Google Scholar
  17. A. Shapiro: Sensitivity analysis of parameterized variational inequalities, Math. Oper. Res. 30(1) (2005) 109–126.https://doi.org/10.1287/moor.1040.0115Google Scholar
  18. Y.-L. Yu: On decomposing the proximal map, Adv. Neural Information Processing Systems 26 (2013) 91–99.Google Scholar

Suggested citation

S. Adly, L. Bourdin, F. Caubet. “On a Decomposition Formula for the Proximal Operator of the Sum of Two Convex Functions.” Journal of Convex Analysis 26 (2019), No. 3, 699–718. https://doi.org/10.68381/jca26037

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.