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An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems

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Abstract

We introduce and investigate the convergence properties of an inertial forward-backward-forward splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitzian operator. By making use of the product space approach, we expand it to the solving of inclusion problems involving mixtures of linearly composed and parallel-sum type monotone operators. We obtain in this way an inertial forward-backward-forward primal-dual splitting algorithm having as main characteristic the fact that in the iterative scheme all operators are accessed separately either via forward or via backward evaluations. We present also the variational case when one is interested in the solving of a primal-dual pair of convex optimization problems with complexly structured objectives, which we also illustrate by numerical experiments in image processing.

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References

  1. Alvarez, F.: On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control. Optim. 38(4), 1102–1119 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14(3), 773–782 (2004)

    Article  MATH  Google Scholar 

  3. Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Attouch, H., Briceño-Arias, L.M., Combettes, P.L.: A parallel splitting method for coupled monotone inclusions. SIAM J. Control. Optim. 48(5), 3246–3270 (2010)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

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

    Book  Google Scholar 

  8. Bot, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin Heidelberg (2010)

    Book  Google Scholar 

  9. Bot, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61(1), 35–65 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bot, R.I., Csetnek, E.R., Heinrich, A.: A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators. SIAM J. Optim. 23(4), 2011–2036 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bot, R.I., Csetnek, E.R., Heinrich, A., Hendrich, C.: On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems. Math. Program. 150(2), 251–279 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472–487 (2015)

    Article  MathSciNet  Google Scholar 

  13. Bot, R.I., Csetnek, E.R., László, S.: An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. arXiv:1410.0641

  14. Bot, R.I., Hendrich, C.: Convergence analysis for a primal-dual monotone + skew splitting algorithm with applications to total variation minimization. J. Math. Imaging Vision 49(3), 551–568 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bot, R.I., Hendrich, C.: A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators. SIAM J. Optim. 23(4), 2541–2565 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Briceño-Arias, L.M., Combettes, P.L.: A monotone + skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 21(4), 1230–1250 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cabot, A., Frankel, P.: Asymptotics for some proximal-like method involving inertia and memory aspects. Set-Valued Var. Anal. 19, 59–74 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20(1–2), 89–97 (2004)

    MathSciNet  Google Scholar 

  19. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision 40(1), 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5-6), 475–504 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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 

  22. Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158 (2), 460–479 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Company, Amsterdam (1976)

    MATH  Google Scholar 

  24. Maingé, P.-E.: Convergence theorems for inertial KM-type algorithms. J. Comput. Appl. Math. 219, 223–236 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Maingé, P.-E., Moudafi, A.: Convergence of new inertial proximal methods for dc programming. SIAM J. Optim. 19(1), 397–413 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155, 447–454 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ochs, P., Chen, Y., Brox, T., Pock, T.: iPiano: Inertial proximal algorithm for non-convex optimization. SIAM J. Imag. Sci. 7(2), 1388–1419 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pesquet, J.-C., Pustelnik, N.: A parallel inertial proximal optimization method. Pac. J. Optim. 8(2), 273–305 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Simons, S.: From Hahn-Banach to Monotonicity. Springer, Berlin (2008)

    MATH  Google Scholar 

  32. Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control. Optim. 29(1), 119–138 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control. Optim. 38(2), 431–446 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Vu, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38(3), 667–681 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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Authors and Affiliations

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

    Radu Ioan Boţ & Ernö Robert Csetnek

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  1. Radu Ioan Boţ
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  2. Ernö Robert Csetnek
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Correspondence to Radu Ioan Boţ.

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Research supported by DFG (German Research Foundation), project BO 2516/4-1.

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Boţ, R.I., Csetnek, E.R. An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer Algor 71, 519–540 (2016). https://doi.org/10.1007/s11075-015-0007-5

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  • DOI: https://doi.org/10.1007/s11075-015-0007-5

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