SpringerLink
Log in

A Projective Splitting Method for Monotone Inclusions: Iteration-Complexity and Application to Composite Optimization

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We propose an inexact projective splitting method to solve the problem of finding a zero of a sum of maximal monotone operators. We perform convergence and complexity analyses of the method by viewing it as a special instance of an inexact proximal point method proposed by Solodov and Svaiter in 2001, for which pointwise and ergodic complexity results have been studied recently by Sicre. Also, for this latter method, we establish convergence rates and complexity bounds for strongly monotone inclusions, from where we obtain linear convergence for our projective splitting method under strong monotonicity and cocoercivity assumptions. We apply the proposed projective splitting scheme to composite convex optimization problems and establish pointwise and ergodic function value convergence rates, extending a recent work of Johnstone and Eckstein.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alotaibi, A., Combettes, P.L., Shahzad, N.: Solving coupled composite monotone inclusions by successive Fejér approximations of their Kuhn–Tucker set. SIAM J. Optim. 24(4), 2076–2095 (2014). https://doi.org/10.1137/130950616

    Article  MathSciNet  MATH  Google Scholar 

  2. Alves, M.M., Monteiro, R.D.C., Svaiter, B.F.: Regularized HPE-type methods for solving monotone inclusions with improved pointwise iteration-complexity bounds. SIAM J. Optim. 26(4), 2730–2743 (2016). https://doi.org/10.1137/15M1038566

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves, M.M., Eckstein, J., Geremia, M.J.G.: Relative-error inertial-relaxed inexact versions of Douglas–Rachford and ADMM splitting algorithms. Comput. Optim. Appl. 75(2), 389–422 (2020). https://doi.org/10.1007/s10589-019-00165-y

    Article  MathSciNet  MATH  Google Scholar 

  4. Bauschke, H.H., Combettes, P.L., Reich, S.: The asymptotic behavior of the composition of two resolvents. Nonlinear Anal. 60, 283–301 (2005). https://doi.org/10.1016/j.na.2004.07.054

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauschke, H.H., Combettes, P.L., Kruk, S.G.: Extrapolation algorithm for affine-convex feasibility problems. Numer. Algorithm 41, 239–274 (2006). https://doi.org/10.1007/s11075-005-9010-6

    Article  MathSciNet  MATH  Google Scholar 

  6. Bauschke, H, H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-48311-5

  7. Bùi, M.N., Combettes, P.L.: Warped proximal iterations for monotone inclusions. J. Math. Anal. Appl. 491(1), 124315 (2020). https://doi.org/10.1016/j.jmaa.2020.124315

    Article  MathSciNet  MATH  Google Scholar 

  8. Burachik, R.S., Iusem, A.N., Svaiter, B.F.: Enlargement of monotone operators with applications to variational inequalities. Set-Valued Anal. 5(2), 159–180 (1997). https://doi.org/10.1023/A:1008615624787

    Article  MathSciNet  MATH  Google Scholar 

  9. Burachik, R.S., Svaiter, B.F.: \(\varepsilon \)-enlargements of maximal monotone operators in Banach spaces. Set-Valued Anal. 7(2), 117–132 (1999). https://doi.org/10.1023/A:1008730230603

    Article  MathSciNet  MATH  Google Scholar 

  10. Burachik, R.S., Sagastizábal, C.A., Scheimberg, S.: An inexact method of partial inverses and a parallel bundle method. Optim. Methods Softw. 21(3), 385–400 (2006). https://doi.org/10.1080/10556780500094887

    Article  MathSciNet  MATH  Google Scholar 

  11. Ceng, L.C., Mordukhovich, B.S., Yao, J.C.: Hybrid approximate proximal method with auxiliary variational inequality for vector optimization. J. Optim. Theory Appl. 146(2), 267–303 (2010). https://doi.org/10.1007/s10957-010-9667-4

    Article  MathSciNet  MATH  Google Scholar 

  12. Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, pp. 185–212 (2011)

  13. Combettes, P.L., Eckstein, J.: Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions. Math. Program. 168(1–2), 645–672 (2018). https://doi.org/10.1007/s10107-016-1044-0

    Article  MathSciNet  MATH  Google Scholar 

  14. Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. (1956). https://doi.org/10.1090/S0002-9947-1956-0084194-4

  15. Eckstein, J., Svaiter, B.F.: A family of projective splitting methods for the sum of two maximal monotone operators. Math. Program. 111(1–2), 173–199 (2008). https://doi.org/10.1007/s10107-006-0070-8

    Article  MathSciNet  MATH  Google Scholar 

  16. Eckstein, J., Svaiter, B.F.: General projective splitting methods for sums of maximal monotone operators. SIAM J. Control. Optim. 48(2), 787–811 (2009). https://doi.org/10.1137/070698816

    Article  MathSciNet  MATH  Google Scholar 

  17. Eckstein, J.: A simplified form of block-iterative operator splitting and an asynchronous algorithm resembling the multi-block alternating direction method of multipliers. J. Optim. Theory Appl. 173, 155–182 (2017). https://doi.org/10.1007/s10957-017-1074-7

    Article  MathSciNet  MATH  Google Scholar 

  18. Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangians: Application to the Numerical Solution of Boundary Value Problems, pp. 299–331. North-Holland, Amsterdam (1983). https://doi.org/10.1016/S0168-2024(08)70034-1

    Chapter  Google Scholar 

  19. Giselsson, P.K.: Nonlinear forward-backward splitting with projection correction. SIAM J. Optim. 31(3), 2199–2226 (2021). ar**v:1908.07449

    Article  MathSciNet  MATH  Google Scholar 

  20. Glowinski, R., Osher, S.J., Yin, W.: Splitting Methods in Communication, Imaging, Science, and Engineering. Springer (Incorporated) (2017)

  21. Johnstone, P.R., Eckstein, J.: Convergence rates for projective splitting. SIAM J. Optim. 29(3), 1931–1957 (2019). https://doi.org/10.1137/18M1203523

  22. Johnstone, P.R., Eckstein, J.: Projective splitting with forward steps: asynchronous and block-iterative operator splitting (2020). ar**v:1803.07043

  23. Johnstone, P.R., Eckstein, J.: Single-forward-step projective splitting: exploiting cocoercivity. Comput. Optim. Appl. 78, 125–166 (2021). https://doi.org/10.1007/s10589-020-00238-3

    Article  MathSciNet  MATH  Google Scholar 

  24. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979). https://doi.org/10.1137/0716071

    Article  MathSciNet  MATH  Google Scholar 

  25. Machado, M.P.: On the complexity of the projective splitting and S**arn’s methods for the sum of two maximal monotone operators. J. Optim. Theory Appl. 178(1), 153–190 (2018). https://doi.org/10.1007/s10957-018-1310-9

    Article  MathSciNet  MATH  Google Scholar 

  26. Machado, M.P.: Projective method of multipliers for linearly constrained convex minimization. Comput. Optim. Appl. 73, 237–273 (2019). https://doi.org/10.1007/s10589-019-00065-1

    Article  MathSciNet  MATH  Google Scholar 

  27. Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Inf. Rech. Opér. 4(Ser. R–3), 154–158 (1970)

    MATH  Google Scholar 

  28. Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962). https://doi.org/10.1215/S0012-7094-62-02933-2

    Article  MathSciNet  MATH  Google Scholar 

  29. Monteiro, R.D.C., Svaiter, B.F.: On the complexity of the hybrid proximal extragradient method for the iterates and the ergodic mean. SIAM J. Optim. 20(6), 2755–2787 (2010). https://doi.org/10.1137/090753127

    Article  MathSciNet  MATH  Google Scholar 

  30. Monteiro, R.D.C., Svaiter, B.F.: Complexity of variants of Tseng’s modified F-B splitting and Korpelevich’s methods for hemivariational inequalities with applications to saddle-point and convex optimization problems. SIAM J. Optim. 21(4), 1688–1720 (2011). https://doi.org/10.1137/100801652

    Article  MathSciNet  MATH  Google Scholar 

  31. Monteiro, R.D.C., Svaiter, B.F.: Iteration-complexity of block-decomposition algorithms and the alternating direction method of multipliers. SIAM J. Optim. 23, 475–507 (2013). https://doi.org/10.1137/110849468

    Article  MathSciNet  MATH  Google Scholar 

  32. Monteiro, R.D.C., Sim, C.: Complexity of the relaxed Peaceman–Rachford splitting method for the sum of two maximal strongly monotone operators. Comput. Optim. Appl. 70, 763–790 (2018). https://doi.org/10.1007/s10589-018-9996-z

    Article  MathSciNet  MATH  Google Scholar 

  33. Moudafi, A.: On the convergence of the forward- backward algorithm for null-point problems. J. Nonlinear Var. Anal. 2, 263–268 (2018). https://doi.org/10.23952/jnva.2.2018.3.01

    Article  MathSciNet  MATH  Google Scholar 

  34. Peaceman, D.W., Rachford, H.H., Jr.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955). https://doi.org/10.1137/0103003

    Article  MathSciNet  MATH  Google Scholar 

  35. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149(1), 75–88 (1970). https://doi.org/10.1090/S0002-9947-1970-0282272-5

    Article  MathSciNet  MATH  Google Scholar 

  36. Rockafellar, R.T.: On the maximal monotonicity of subdifferential map**s. Pac. J. Math. 33, 209–216 (1970). https://doi.org/10.2140/pjm.1970.33.209

    Article  MathSciNet  MATH  Google Scholar 

  37. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976). https://doi.org/10.1137/0314056

    Article  MathSciNet  MATH  Google Scholar 

  38. Sicre, M.R.: On the complexity of a hybrid proximal extragradient projective method for solving monotone inclusion problems. Optim. Appl., Comput. (2020). https://doi.org/10.1007/s10589-020-00200-39

  39. Simons, S.: Minimax and Monotonicity. Springer, Berlin, Heidelberg (1998)

    Book  MATH  Google Scholar 

  40. Solodov, M.V., Svaiter, B.F.: A hybrid projection-proximal point algorithm. J. Convex Anal. 6(1), 59–70 (1999)

    MathSciNet  MATH  Google Scholar 

  41. Solodov, M.V., Svaiter, B.F.: A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Anal. 7(4), 323–345 (1999). https://doi.org/10.1023/A:1008777829180

    Article  MathSciNet  MATH  Google Scholar 

  42. Solodov, M.V., Svaiter, B.F.: A unified framework for some inexact proximal point algorithms. Numer. Funct. Anal. Optim. 22(7–8), 1013–1035 (2001). https://doi.org/10.1081/NFA-100108320

    Article  MathSciNet  MATH  Google Scholar 

  43. Tseng, P.: A modified forward-backward splitting method for maximal monotone map**s. SIAM J. Control Optim. 38, 431–446 (2000). https://doi.org/10.1137/S0363012998338806

    Article  MathSciNet  MATH  Google Scholar 

  44. Villa, S., Salzo, S., Baldassarre, L., Verri, A.: Accelerated and inexact forward-backward algorithms. J. SIAM J. Optim. 23(3), 1607–1633 (2013). https://doi.org/10.1137/110844805

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal da Bahia, Campus de Ondina, Av. Adhemar de Barros, S/N Ondina, Salvador, BA, 40170-110, Brazil

    Majela Pentón Machado & Mauricio Romero Sicre

Authors
  1. Majela Pentón Machado
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mauricio Romero Sicre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Majela Pentón Machado.

Additional information

Communicated by Aviv Gibali.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pentón Machado, M., Sicre, M.R. A Projective Splitting Method for Monotone Inclusions: Iteration-Complexity and Application to Composite Optimization. J Optim Theory Appl 198, 552–587 (2023). https://doi.org/10.1007/s10957-023-02214-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02214-3

Keywords

Mathematics Subject Classification

Search

Navigation