Skip to main content
SpringerLink
Log in

On a Decomposition Formula for the Resolvent Operator of the Sum of Two Set-Valued Maps with Monotonicity Assumptions

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

The aim of the present work is to provide an explicit decomposition formula for the resolvent operator \(\mathrm {J}_{A+B}\) of the sum of two set-valued maps A and B in a Hilbert space. For this purpose we introduce a new operator, called the A-resolvent operator of B and denoted by \(\mathrm {J}^A_B\), which generalizes the usual notion. Then, our main result lies in the decomposition formula \(\mathrm {J}_{A+B}=\mathrm {J}_A\circ \mathrm {J}^A_B\) holding true when A is monotone. Several properties of \(\mathrm {J}^A_B\) are deeply investigated in this paper. In particular the relationship between \(\mathrm {J}^A_B\) and an extended version of the classical Douglas–Rachford operator is established, which allows us to propose a weakly convergent algorithm that computes numerically \(\mathrm {J}^A_B\) (and thus \(\mathrm {J}_{A+B}\) from the decomposition formula) when A and B are maximal monotone. In order to illustrate our theoretical results, we give an application in elliptic PDEs. Precisely the decomposition formula is used to point out the relationship between the classical obstacle problem and a new nonlinear PDE involving a partially blinded elliptic operator. Some numerical experiments, using the finite element method, are carried out in order to support our approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Adly, S., Bourdin, L., Caubet, F.: On a decomposition formula for the proximity operator of the sum of two convex functions. J. Convex Anal. 26(3), 699–718 (2019)

    MATH  Google Scholar 

  2. Aragón Artacho, F.J., Campoy, R.: Computing the resolvent of the sum of maximally monotone operators with the averaged alternating modified reflections algorithm. J Optim Theory Appl. 181(3), 709–726 (2019)

    Article  MathSciNet  Google Scholar 

  3. Aubin, J.-P., Frankowska, H.: Set-valued analysis. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, (2009). Reprint of the 1990 edition [MR1048347]

  4. Bauschke, H.H., Combettes, P.I.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 2nd edn. Springer, New York (2017)

    Google Scholar 

  5. Borwein, J.M.: Fifty years of maximal monotonicity. Optim. Lett. 4(4), 473–490 (2010)

    Article  MathSciNet  Google Scholar 

  6. Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50) (1973)

  7. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)

    MATH  Google Scholar 

  8. Brezis, H.R., Stampacchia, G.: Sur la régularité de la solution d’inéquations elliptiques. Bull. Soc. Math. France 96, 153–180 (1968)

    Article  MathSciNet  Google Scholar 

  9. Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer Optimization and Its Applications, vol. 8. Springer, New York (2008)

    Google Scholar 

  10. Combettes, P.L., Dũng, D., Vũ, B.C.: Dualization of signal recovery problems. Set-Valued Var. Anal. 18(3–4), 373–404 (2010)

    Article  MathSciNet  Google Scholar 

  11. Douglas Jr., J., Rachford Jr., H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)

    Article  MathSciNet  Google Scholar 

  12. Duvaut, G., Lions, J.-L.: Inequalities in mechanics and physics. Springer-Verlag, Berlin-New York. Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, p. 219 (1976)

    Book  Google Scholar 

  13. Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)

    Article  MathSciNet  Google Scholar 

  14. Elliott, C.M., Ockendon, J.R.: Weak and Variational Methods for Moving Boundary Problems. Research Notes in Mathematics, vol. 59. Pitman (Advanced Publishing Program), Boston, London (1982)

    MATH  Google Scholar 

  15. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence, RI (2010)

    MATH  Google Scholar 

  16. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Scientific Computation. Springer-Verlag, Berlin, (2008). Reprint of the 1984 original

  17. Lions, P.-L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)

    Article  MathSciNet  Google Scholar 

  18. Martinet, B.: Détermination approchée d’un point fixe d’une application pseudo-contractante. Cas de l’application prox. C. R. Acad. Sci. Paris Sér. A-B, 274:A163–A165 (1972)

  19. Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)

    Article  MathSciNet  Google Scholar 

  20. Minty, G.J.: On some aspects of the theory of monotone operators. In: Theory and Applications of Monotone Operators (Proc. NATO Advanced Study Inst., Venice, 1968), pp. 67–82. Edizioni “Oderisi”, Gubbio (1969)

  21. Moreau, J.-J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)

    Article  MathSciNet  Google Scholar 

  22. Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, vol. 1364, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  23. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

    Article  MathSciNet  Google Scholar 

  24. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)

    Article  MathSciNet  Google Scholar 

  25. Simons, S., Zalinescu, C.: A new proof for Rockafellar’s characterization of maximal monotone operators. Proc. Am. Math. Soc. 132(10), 2969–2972 (2004)

    Article  MathSciNet  Google Scholar 

  26. Steinbach, J.: A Variational Inequality Approach to Free Boundary Problems with Applications in Mould Filling. International Series of Numerical Mathematics, vol. 136. Birkhäuser Verlag, Basel (2002)

    Book  Google Scholar 

  27. Zeidler, E.: Nonlinear functional analysis and its applications. II/B. Springer-Verlag, New York. Nonlinear monotone operators. Translated from the German by the author and Leo F. Boron (1990)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Recherche XLIM, UMR CNRS 7252, Université de Limoges, 123, Avenue Albert Thomas, 87060, Limoges Cedex, France

    Samir Adly & Loïc Bourdin

Authors
  1. Samir Adly
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Loïc Bourdin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samir Adly.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Adly, S., Bourdin, L. On a Decomposition Formula for the Resolvent Operator of the Sum of Two Set-Valued Maps with Monotonicity Assumptions. Appl Math Optim 80, 715–732 (2019). https://doi.org/10.1007/s00245-019-09599-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-019-09599-6

Keywords

Mathematics Subject Classification

Search

Navigation