Skip to main content
SpringerLink
Log in

Subgradient Projectors: Extensions, Theory, and Characterizations

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

Subgradient projectors play an important role in optimization and for solving convex feasibility problems. For every locally Lipschitz function, we can define a subgradient projector via generalized subgradients even if the function is not convex. The paper consists of three parts. In the first part, we study basic properties of subgradient projectors and give characterizations when a subgradient projector is a cutter, a local cutter, or a quasi-nonexpansive mapping. We present global and local convergence analyses of subgradent projectors. Many examples are provided to illustrate the theory. In the second part, we investigate the relationship between the subgradient projector of a prox-regular function and the subgradient projector of its Moreau envelope. We also characterize when a mapping is the subgradient projector of a convex function. In the third part, we focus on linearity properties of subgradient projectors. We show that, under appropriate conditions, a linear operator is a subgradient projector of a convex function if and only if it is a convex combination of the identity operator and a projection operator onto a subspace. In general, neither a convex combination nor a composition of subgradient projectors of convex functions is a subgradient projector of a convex function.

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

Similar content being viewed by others

References

  1. Adly, S., Nacry, F., Thibault, L.: Preservation of prox-regularity of sets with applications to constrained optimization. SIAM J. Optim. 26(1), 448–473 (2016)

    Article  MathSciNet  Google Scholar 

  2. Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: functional characterizations and related concepts. Trans. Amer. Math. Soc. 357(4), 1275–1301 (2005)

    Article  MathSciNet  Google Scholar 

  3. Bacák, M., Borwein, J.M., Eberhard, A., Mordukhovich, B.S.: Infimal convolutions and Lipschitzian properties of subdifferentials for prox-regular functions in Hilbert spaces. J. Convex Anal. 17(3-4), 737–763 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Baillon, J.B., Bruck, R.R., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4, 1–9 (1978)

    MathSciNet  MATH  Google Scholar 

  5. Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Bregman monotone optimization algorithms. SIAM J. Control Optim. 42, 596–636 (2003)

    Article  MathSciNet  Google Scholar 

  6. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer (2011)

  7. Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejer-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)

    Article  MathSciNet  Google Scholar 

  8. Bauschke, H.H., Combettes, P.L., Noll, D.: Joint minimization with alternating Bregman proximity operators. Pac. J. Optim. 2(3), 401–424 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Bauschke, H.H., Chen, J., Wang, X.: A Bregman projection method for approximating fixed points of quasi-Bregman nonexpansive mappings. Appl. Anal. 94, 75–84 (2015)

    Article  MathSciNet  Google Scholar 

  10. Bauschke, H.H., Wang, C., Wang, X., Xu, J.: On subgradient projectors. SIAM J. Optim. 25(2), 1064–1082 (2015)

    Article  MathSciNet  Google Scholar 

  11. Bauschke, H.H., Wang, C., Wang, X., Xu, J.: On the finite convergence of a projected cutter method. J. Optim. Theory Appl. 165(3), 901–916 (2015)

    Article  MathSciNet  Google Scholar 

  12. Bauschke, H.H., Wang, X., Yao, L.: Monotone linear relations: maximality and Fitzpatrick functions. J. Convex Anal. 16, 673–686 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Bernard, F., Thibault, L.: Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303(1), 1–14 (2005)

    Article  MathSciNet  Google Scholar 

  14. Bernard, F., Thibault, L.: Prox-regularity of functions and sets in Banach spaces. Set-Valued Anal. 12(1-2), 25–47 (2004)

    Article  MathSciNet  Google Scholar 

  15. Borwein, J.M., Moors, W.B.: Essentially smooth Lipschitz functions. J. Funct. Anal. 149(2), 305–351 (1997)

    Article  MathSciNet  Google Scholar 

  16. Borwein, J.M., Moors, W.B., Wang, X.: Generalized subdifferentials: a Baire categorical approach. Trans. Amer. Math. Soc. 353(10), 3875–3893 (2001)

    Article  MathSciNet  Google Scholar 

  17. Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples, 2nd edn. Springer, New York (2006)

    Book  Google Scholar 

  18. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer-Verlag, New York (2005)

    MATH  Google Scholar 

  19. Capricelli, T.D.: Algorithmes de Projections Convexes Généralisées et Applications en Imagerie Médicale, Ph.D. dissertation. University Pierre & Marie Curie (Paris 6), Paris (2008)

    Google Scholar 

  20. Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, 2057. Springer, Heidelberg (2012)

    Google Scholar 

  21. Censor, Y., Chen, W., Pajoohesh, H.: Finite convergence of a subgradient projections method with expanding controls. Appl. Math. Optim. 64(2), 273–285 (2011)

    Article  MathSciNet  Google Scholar 

  22. Censor, Y., Lent, A.: Cyclic subgradient projections. Math. Programming 24(1), 233–235 (1982)

    Article  MathSciNet  Google Scholar 

  23. Clarke, F.H.: Optimization and Nonsmooth Analysis, Second Edition Classics in Applied Mathematics, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990)

  24. Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections. IEEE Trans. Image Process. 6(4), 493–506 (1997)

    Article  Google Scholar 

  25. Combettes, P.L., Luo, J.: An adaptive level set method for nondifferentiable constrained image recovery. IEEE Trans. Image Process. 11(11), 1295–1304 (2002)

    Article  MathSciNet  Google Scholar 

  26. Daniilidis, A., Georgiev, P.: Approximate convexity and submonotonicity. J. Math. Anal. Appl. 291(1), 292–301 (2004)

    Article  MathSciNet  Google Scholar 

  27. Deutsch, F.: Best Approximation in Inner Product Spaces. Springer (2001)

  28. Evans, L. C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton FL (1992)

    MATH  Google Scholar 

  29. Fukushima, M.: A finitely convergent algorithm for convex inequalities. IEEE Trans. Automat. Control 27(5), 1126–1127 (1982)

    Article  MathSciNet  Google Scholar 

  30. Hesse, R., Luke, D.R.: Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. SIAM J. Optim. 23(4), 2397–2419 (2013)

    Article  MathSciNet  Google Scholar 

  31. Ioffe, A.D.: Approximate subdifferentials and applications, I: the finite-dimensional theory. Trans. Amer. Math. Soc. 281(1), 389–416 (1984)

    MathSciNet  MATH  Google Scholar 

  32. Jourani, A., Thibault, L., Zagrodny, D.: Differential properties of the Moreau envelope. J. Funct. Anal. 266(3), 1185–1237 (2014)

    Article  MathSciNet  Google Scholar 

  33. Kan, C., Song, W.: The Moreau envelope function and proximal mapping in the sense of the Bregman distance. Nonlinear Anal. 75(3), 1385–1399 (2012)

    Article  MathSciNet  Google Scholar 

  34. Kiwiel, K.C.: The efficiency of subgradient projection methods for convex optimization. I. General level methods. SIAM J. Control Optim. 34(2), 660–676 (1996)

    Article  MathSciNet  Google Scholar 

  35. Kiwiel, K.C.: The efficiency of subgradient projection methods for convex optimization. II. Implementations and extensions. SIAM J. Control Optim. 34(2), 677–697 (1996)

    Article  MathSciNet  Google Scholar 

  36. Kiwiel, K.C.: A Bregman-projected subgradient method for convex constrained nondifferentiable minimization. In: Operations Research Proceedings 1996 (Braunschweig), pp 26–30. Springer, Berlin (1997)

    MATH  Google Scholar 

  37. Luke, D.R., Thao, N.H., Tam, M.K.: Quantitative convergence analysis of iterated expansive, set-valued mappings. arXiv:1605.05725 (2016)

  38. Meyer, C.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000)

  39. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer-Verlag (2006)

  40. Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348, 1235–1280 (1996)

    Article  MathSciNet  Google Scholar 

  41. Ogura, N., Yamada, I.: A deep outer approximating half space of the level set of certain quadratic functions. J. Nonlinear Convex Anal. 6(1), 187–201 (2005)

    MathSciNet  MATH  Google Scholar 

  42. Pang, C.H.: Finitely convergent algorithm for nonconvex inequality problems, arXiv:1405.7280 (2014)

  43. Pauwels, B.: Subgradient projection operators, arXiv:1403.7237v1 (2014)

  44. Poliquin, R.A., Rockafellar, R.T.: Prox-regular functions in variational analysis. Trans. Amer. Math. Soc. 348(5), 1805–1838 (1996)

    Article  MathSciNet  Google Scholar 

  45. Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9, 14–29 (1969). (The original version appeared in Akademija Nauk SSSR. žurnal Vyčislitel’ noı̆ Matematiki i Matematičeskoı̆ Fiziki 9 (1969), 509–521.)

    Article  Google Scholar 

  46. Polyak, B.T.: Introduction to Optimization. Optimization Software (1987)

  47. Polyak, B.T.: Random algorithms for solving convex inequalities Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 409–422. Elsevier (2001)

    Google Scholar 

  48. Richardson, L.F.: Measure and Integration: A Concise Introduction to Real Analysis. Wiley (2009)

  49. Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)

  50. Rockafellar, R.T., Wets, R.: Variational Analysis. Springer corrected 3rd printing (2009)

  51. Spingarn, J.E.: Submonotone subdifferentials of Lipschitz functions. Trans. Amer. Math. Soc. 264(1), 77–89 (1981)

    Article  MathSciNet  Google Scholar 

  52. Van Ngai, H., Luc, D.T., Théra, M.: Approximate convex functions. J. Nonlinear Convex Anal. 1(2), 155–176 (2000)

    MathSciNet  MATH  Google Scholar 

  53. Wang, X.: Subdifferentiability of real functions. Real Anal. Exchange 30(1), 137–171 (2004/05)

    Article  MathSciNet  Google Scholar 

  54. Xu, J.: Subgradient Projectors: Theory, Extensions, and Algorithms. Ph.D. dissertation, University of British Columbia, Kelowna (2016)

    Google Scholar 

  55. Yamada, I., Ogura, N.: Adaptive projected subgradient method for asymptotic minimization of sequence of nonnegative convex functions. Numer. Funct. Anal. Optim. 25(7-8), 593–617 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank two anonymous referees for careful reading and constructive suggestions on the paper. HHB was partially supported by a Discovery Grant of the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Canada Research Chair Program. CW was partially supported by National Natural Science Foundation of China (11401372). XW was partially supported by a Discovery Grant of the Natural Sciences and Engineering Research Council of Canada (NSERC). JX was supported by by NSERC grants of HHB and XW.

Author information

Authors and Affiliations

  1. Mathematics, University of British Columbia, Kelowna, BC, V1V 1V7, Canada

    Heinz H. Bauschke, Xianfu Wang & Jia Xu

  2. Department of Mathematics, Shanghai Maritime University, Shanghai, China

    Caifang Wang

Authors
  1. Heinz H. Bauschke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Caifang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xianfu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jia Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Heinz H. Bauschke.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bauschke, H.H., Wang, C., Wang, X. et al. Subgradient Projectors: Extensions, Theory, and Characterizations. Set-Valued Var. Anal 26, 1009–1078 (2018). https://doi.org/10.1007/s11228-017-0415-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-017-0415-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation