A Projected Primal–Dual Method for Solving Constrained Monotone Inclusions

Abstract

In this paper, we provide an algorithm for solving constrained composite primal–dual monotone inclusions, i.e., monotone inclusions in which a priori information on primal–dual solutions is represented via closed and convex sets. The proposed algorithm incorporates a projection step onto the a priori information sets and generalizes methods proposed in the literature for solving monotone inclusions. Moreover, under the presence of strong monotonicity, we derive an accelerated scheme inspired on the primal–dual algorithm applied to the more general context of constrained monotone inclusions. In the particular case of convex optimization, our algorithm generalizes several primal–dual optimization methods by allowing a priori information on solutions. In addition, we provide an accelerated scheme under strong convexity. An application of our approach with a priori information is constrained convex optimization problems, in which available primal–dual methods impose constraints via Lagrange multiplier updates, usually leading to slow algorithms with unfeasible primal iterates. The proposed modification forces primal iterates to satisfy a selection of constraints onto which we can project, obtaining a faster method as numerical examples exhibit. The obtained results extend and improve several results in the literature.

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References

  1. 1.

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

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled convex minimization problems-applications to dynamical games and PDE’s. J. Convex Anal. 15, 485–506 (2008)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Attouch, H., Briceño-Arias, L.M., Combettes, P.L.: A strongly convergent primal–dual method for nonoverlapping domain decomposition. Numer. Math. 133, 433–470 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Briceño-Arias, L.M., Kalise, D., Silva, F.J.: Proximal methods for stationary Mean Field Games with local couplings. SIAM J. Control Optim. 56(2), 801–836 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    Google Scholar 

  6. 6.

    Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, pp. 299–331. North-Holland, Amsterdam (1983)

    Google Scholar 

  7. 7.

    Mercier, B.: Topics in Finite Element Solution of Elliptic Problems. Lectures on Mathematics, vol. 63. Tata Institute of Fundamental Research, Bombay (1979)

    Google Scholar 

  8. 8.

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

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Briceño-Arias, L.M., Davis, D.: Forward-backward-half forward algorithm for solving monotone inclusions. SIAM J. Optim. 28, 2839–2871 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

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

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

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

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

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

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Esser, E., Zhang, X., Chan, T.F.: A general framework for a class of first order primal–dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3, 1015–1046 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    He, B., Yuan, X.: Convergence analysis of primal–dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5, 119–149 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Chambolle, A., Pock, T.: On the ergodic convergence rates of a first-order primal–dual algorithm. Math. Program. 159, 253–287 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Lorenz, D.A., Pock, T.: An inertial forward–backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Briceño-Arias, L.M., Kalise, D., Kobeissi, Z., Laurière, M., González, A.M., Silva, F.J.: On the implementation of a primal–dual algorithm for second order time-dependent mean field games with local couplings. arXiv:1802.07902

  18. 18.

    Boţ, 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, 251–279 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)

    Google Scholar 

  20. 20.

    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, 307–330 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

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

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Malitsky, Y., Pock, T.: A first-order primal–dual algorithm with linesearch. SIAM J. Optim. 28, 411–432 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

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

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Boţ, R.I., Csetnek, E.R.: An inertial forward–backward–forward primal–dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71, 519–540 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

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

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The authors thank the two anonymous referees which significantly helped to improve the quality of this manuscript. In addition, the authors thank the “Programa de financiamiento basal” from CMM–Universidad de Chile and the project DGIP-UTFSM PI-M-18.14 from Universidad Técnica Federico Santa María.

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Correspondence to Luis Briceño-Arias.

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Communicated by Juan-Enrique Martínez Legaz.

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Briceño-Arias, L., López Rivera, S. A Projected Primal–Dual Method for Solving Constrained Monotone Inclusions. J Optim Theory Appl 180, 907–924 (2019). https://doi.org/10.1007/s10957-018-1430-2

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Keywords

  • Accelerated schemes
  • Constrained convex optimization
  • Monotone operator theory
  • Proximity operator
  • Splitting algorithms

Mathematics Subject Classification

  • 47H05
  • 65K05
  • 65K15
  • 90C25