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Backward Penalty Schemes for Monotone Inclusion Problems

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Abstract

In this paper, we are concerned with solving monotone inclusion problems expressed by the sum of a set-valued maximally monotone operator with a single-valued maximally monotone one and the normal cone to the nonempty set of zeros of another set-valued maximally monotone operator. Depending on the nature of the single-valued operator, we propose two iterative penalty schemes, both addressing the set-valued operators via backward steps. The single-valued operator is evaluated via a single forward step if it is cocoercive, and via two forward steps if it is monotone and Lipschitz continuous. The latter situation represents the starting point for dealing with complexly structured monotone inclusion problems from algorithmic point of view.

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References

  1. Attouch, H., Czarnecki, M.-O.: Asymptotic behavior of coupled dynamical systems with multiscale aspects. J. Differ. Equ. 248(6), 1315–1344 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Attouch, H., Czarnecki, M.-O., Peypouquet, J.: Prox-penalization and splitting methods for constrained variational problems. SIAM J. Optim. 21(1), 149–173 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Attouch, H., Czarnecki, M.-O., Peypouquet, J.: Coupling forward-backward with penalty schemes and parallel splitting for constrained variational inequalities. SIAM J. Optim. 21(4), 1251–1274 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Peypouquet, J.: Coupling the gradient method with a general exterior penalization scheme for convex minimization. J. Optim. Theory Appl. 153(1), 123–138 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Noun, N., Peypouquet, J.: Forward-backward penalty scheme for constrained convex minimization without inf-compactness. J. Optim. Theory Appl. 158(3), 787–795 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boţ, R.I., Csetnek, E.R.: Forward-backward and Tseng’s type penalty schemes for monotone inclusion problems. Set-Valued Var. Anal. 22(2), 313–331 (2013)

    Google Scholar 

  7. Boţ, R.I., Csetnek, E.R.: A Tseng’s type penalty scheme for solving inclusion problems involving linearly composed and parallel-sum type monotone operators. Vietnam J. Math. 42(4), 451–465 (2014)

    Article  MathSciNet  Google Scholar 

  8. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011)

  9. Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  10. Boţ, R. I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer (2010)

  11. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1999)

    Book  Google Scholar 

  12. Simons, S.: From Hahn-Banach to Monotonicity. Lecture Notes in Mathematics, vol. 1693. Springer (2008)

  13. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  14. Bauschke, H.H., McLaren, D.A., Sendov, H.S.: Fitzpatrick functions: inequalities, examples and remarks on a problem by S. Fitzpatrick. J. Convex Anal. 13(3–4), 499–523 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13(3–4), 561–586 (2006)

    MathSciNet  MATH  Google Scholar 

  16. Boţ, R.I., Csetnek, E.R.: An application of the bivariate inf-convolution formula to enlargements of monotone operators. Set-Valued Anal. 16(7–8), 983–997 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Burachik, R.S., Svaiter, B.F.: Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal. 10(4), 297–316 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fitzpatrick, S.: Representing monotone operators by convex functions. Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988). In: Proceedings of the Centre for Mathematical Analysis, vol. 20, pp. 59–65 (1988)

  19. Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33(1), 209–216 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  20. Baillon, J.-B., Haddad, G.: Quelques propriétés des opérateurs angle-bornés et \(n\)-cycliquement monotones. Isr. J. Math. 26(2), 137–150 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  21. Attouch, H., Théra, M.: A general duality principle for the sum of two operators. J. Convex Anal. 3(1), 1–24 (1996)

    MathSciNet  MATH  Google Scholar 

  22. Combettes, P.L., Pesquet, J.-C.: Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal. 20(2), 307–330 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. Springer Monographs in Mathematics. Springer, Dordrecht (2012)

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Acknowledgments

The authors thank the anonymous referees for several improvements in the paper.

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Authors and Affiliations

  1. University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Sebastian Banert & Radu Ioan Boţ

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  1. Sebastian Banert
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  2. Radu Ioan Boţ
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Correspondence to Radu Ioan Boţ.

Additional information

The research of R.I. Boţ was partially supported by the German Research Foundation (DFG), Project BO 2516/4-1.

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Banert, S., Boţ, R.I. Backward Penalty Schemes for Monotone Inclusion Problems. J Optim Theory Appl 166, 930–948 (2015). https://doi.org/10.1007/s10957-014-0700-x

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