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Forward–Partial Inverse–Forward Splitting for Solving Monotone Inclusions

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Abstract

In this paper, we provide a splitting method for finding a zero of the sum of a maximally monotone operator, a Lipschitzian monotone operator, and a normal cone to a closed vector subspace of a real Hilbert space. The problem is characterised by a simpler monotone inclusion involving only two operators: the partial inverse of the maximally monotone operator with respect to the vector subspace and a suitable Lipschitzian monotone operator. By applying the Tseng’s method in this context, we obtain a fully split algorithm that exploits the whole structure of the original problem and generalises partial inverse and Tseng’s methods. Connections with other methods available in the literature are provided, and the flexibility of our setting is illustrated via applications to some inclusions involving \(m\) maximally monotone operators, to primal-dual composite monotone inclusions, and to zero-sum games.

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References

  1. Combettes, P.L., Pesquet, J.-C.: A proximal decomposition method for solving convex variational inverse problems. Inverse Problems 24, 065014 (2008)

  2. Spingarn, J.E.: Applications of the method of partial inverses to convex programming: decomposition. Math. Program. 32, 199–223 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lions, J.-L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. 20, 493–519 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  4. Tseng, P.: Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming. Math. Program. 48, 249–263 (1990)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lions, P.-L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Spingarn, J.E.: Partial inverse of a monotone operator. Appl. Math. Optim. 10, 247–265 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mercier, B.: Inéquations Variationnelles de la Mécanique. Publications Mathématiques d’Orsay, no. 80.01, Université de Paris-XI, Orsay, France (1980)

  11. Zeidler, E.: Nonlinear Functional Analysis and Its Applications II/B—Nonlinear Monotone Operators. Springer, New York (1990)

    Book  MATH  Google Scholar 

  12. Jofré, A., Rockafellar, R.T., Wets, R.J.-B.: Variational inequalities and economic equilibrium. Math. Oper. Res. 32, 32–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pennanen, T.: Introduction to convex optimization in financial markets. Math. Program. 134, 157–186 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Aujol, J.-F., Aubert, G., Blanc-Feraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vision 22, 71–88 (2005)

  15. Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57, 1413–1457 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  18. Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs 49. Amer Math. Soc., Providence, RI (1997)

  19. Bertsekas, D.P., Gafni, E.M.: Projection methods for variational inequalities with application to the traffic assignment problem. Math. Program. Stud. 17, 139–159 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fukushima, M.: The primal Douglas-Rachford splitting algorithm for a class of monotone mappings with applications to the traffic equilibrium problem. Math. Program. 72, 1–15 (1996)

    MathSciNet  MATH  Google Scholar 

  21. Gilpin, A., Peña, J., Sandholm, T.: First-order algorithm with \({\cal O}(\ln (1/\epsilon ))\) convergence for \(\epsilon \)-equilibrium in two-person zero-sum games. Math. Program. Ser. A 133, 279–298 (2012)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Briceño-Arias, L.M.: Forward-Douglas-Rachford splitting and forward-partial inverse method for solving monotone inclusions. Optimization (2013). doi:10.1080/02331934.2013.855210

  25. Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. Combettes, P.L.: Iterative construction of the resolvent of a sum of maximal monotone operators. J. Convex Anal. 16, 727–748 (2009)

    MathSciNet  MATH  Google Scholar 

  27. 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Svaiter, B.F.: Weak convergence on Douglas-Rachford method. SIAM J. Control Optim. 49, 280–287 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Briceño-Arias, L.M.: Outer approximation method for constrained composite fixed point problems involving Lipschitz pseudo contractive operators. Numer. Funct. Anal. Optim. 32, 1099–1115 (2011)

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

    Book  MATH  Google Scholar 

  31. 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)

    Article  MATH  Google Scholar 

  32. Luke, D.R.: Finding best approximation pairs relative to a convex and prox-regular set in a Hilbert space. SIAM J. Optim. 19, 714–739 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Combettes, P.L.: A primal-dual method of partial inverses for composite inclusions. Optim. Lett. published on-line (2014)

  34. Raguet, H., Fadili, J., Peyré, G.: Generalized forward–backward splitting. SIAM J. Imaging Sci. 6, 1199–1226 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Boţ, R.I., László, S.: On the generalized parallel sum of two maximal monotone operators of Gossez type (D). J. Math. Anal. Appl. 391, 82–98 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Attouch, H., Redont, P., Soubeyran, A.: A new class of alternating proximal minimization algorithms with costs-to-move. SIAM J. Optim. 18, 1061–1081 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. 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 

  40. Briceño-Arias, L.M., Combettes, P.L.: Monotone operator methods for Nash equilibria in non-potential games. In: Bailey, D., Bauschke, H.H., Borwein, P., Garvan, F., Théra, M., Vanderwerff, J., Wolkowicz, H. (eds.) Computational and Analytical Mathematics vol. 50, pp. 143–159. Springer, New York (2013)

    Chapter  Google Scholar 

  41. Rockafellar, R.T.: Monotone operators associated with saddle-functions and minimax problems. In: Browder, F.E. (ed.) Nonlinear Functional Analysis Part 1, Proceedings of Symposium Pure Math., vol. 18, pp. 241–250. Amer. Math. Soc., Providence, RI (1970)

  42. Weibull, J.W.: Evolutionary Game Theory. MIT Press, Cambridge (1995)

    MATH  Google Scholar 

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Acknowledgments

The author is thankful to the anonymous reviewers for comments and suggestions, which improved the quality of the paper. This work was supported by CONICYT under Grants FONDECYT 3120054, FONDECYT 11140360, ECOS-CONICYT C13E03, Anillo ACT 1106, Math-Amsud N 13MATH01 and by “Programa de financiamiento basal” from the Center for Mathematical Modeling, Universidad de Chile.

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  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Av. Vicuña Mackenna 3939, San Joaquín, Santiago, Chile

    Luis M. Briceño-Arias

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  1. Luis M. Briceño-Arias
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Correspondence to Luis M. Briceño-Arias.

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Communicated by Viorel Barbu.

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Briceño-Arias, L.M. Forward–Partial Inverse–Forward Splitting for Solving Monotone Inclusions. J Optim Theory Appl 166, 391–413 (2015). https://doi.org/10.1007/s10957-015-0703-2

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