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Maximal Monotone Operators and the Proximal Point Algorithm in the Presence of Computational Errors

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Abstract

In a finite-dimensional Euclidean space, we study the convergence of a proximal point method to a solution of the inclusion induced by a maximal monotone operator, under the presence of computational errors. Most results known in the literature establish the convergence of proximal point methods, when computational errors are summable. In the present paper, the convergence of the method is established for nonsummable computational errors. We show that the proximal point method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauschke, H.H., Goebel, R., Lucet, Y., Wang, X.: The proximal average: basic theory. SIAM J. Optim. 19, 766–785 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burachik, R.S., Lopes, J.O., Da Silva, G.J.P.: An inexact interior point proximal method for the variational inequality. Comput. Appl. Math. 28, 15–36 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Butnariu, D., Kassay, G.: A proximal-projection method for finding zeros of set-valued operators. SIAM J. Control Optim. 47, 2096–2136 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Censor, Y., Zenios, S.A.: The proximal minimization algorithm with D-functions. J. Optim. Theory Appl. 73, 451–464 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)

    Article  MathSciNet  Google Scholar 

  7. Hager, W.W., Zhang, H.: Asymptotic convergence analysis of a new class of proximal point methods. SIAM J. Control Optim. 46, 1683–1704 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kaplan, A., Tichatschke, R.: Bregman-like functions and proximal methods for variational problems with nonlinear constraints. Optimization 56, 253–265 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kassay, G.: The proximal points algorithm for reflexive Banach spaces. Stud. Univ. Babes-Bolyai Math. 30, 9–17 (1985)

    MathSciNet  MATH  Google Scholar 

  10. Martinet, B.: Pertubation des methodes d’optimisation: application. RAIRO. Anal. Numér. 12, 153–171 (1978)

    MathSciNet  MATH  Google Scholar 

  11. Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  12. Minty, G.J.: On the monotonicity of the gradient of a convex function. Pac. J. Math. 14, 243–247 (1964)

    MathSciNet  MATH  Google Scholar 

  13. Moreau, J.J.: Proximite et dualite dans un espace Hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)

    MathSciNet  MATH  Google Scholar 

  14. Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Solodov, M.V., Svaiter, B.F.: Error bounds for proximal point subproblems and associated inexact proximal point algorithms. Math. Program. 88, 371–389 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Solodov, M.V., Svaiter, B.F.: A unified framework for some inexact proximal point algorithms. Numer. Funct. Anal. Optim. 22, 1013–1035 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xu, H.-K.: A regularization method for the proximal point algorithm. J. Glob. Optim. 36, 115–125 (2006)

    Article  MATH  Google Scholar 

  19. Yamashita, N., Kanzow, C., Morimoto, T., Fukushima, M.: An infeasible interior proximal method for convex programming problems with linear constraints. J. Nonlinear Convex Anal. 2, 139–156 (2001)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Technion, Haifa, Israel

    A. J. Zaslavski

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  1. A. J. Zaslavski
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Correspondence to A. J. Zaslavski.

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Communicated by V.F. Demyanov.

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Zaslavski, A.J. Maximal Monotone Operators and the Proximal Point Algorithm in the Presence of Computational Errors. J Optim Theory Appl 150, 20–32 (2011). https://doi.org/10.1007/s10957-011-9820-8

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  • DOI: https://doi.org/10.1007/s10957-011-9820-8

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