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Strong Convergence of Projected Subgradient Methods for Nonsmooth and Nonstrictly Convex Minimization

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In this paper, we establish a strong convergence theorem regarding a regularized variant of the projected subgradient method for nonsmooth, nonstrictly convex minimization in real Hilbert spaces. Only one projection step is needed per iteration and the involved stepsizes are controlled so that the algorithm is of practical interest. To this aim, we develop new techniques of analysis which can be adapted to many other non-Fejérian methods.

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  1. 1.

    Albert, Y.I.: Recurrence relations and variational inequalities. Soviet Mathematics, Doklady, 27, 511–517 (1983)

  2. 2.

    Albert, Y.I., Iusem, A.N.: Extension of subgradient techniques for nonsmooth optimization in a Banach space. Set-Valued Anal. 9, 315–335 (2001)

  3. 3.

    Albert, Y.I., Iusem, A.N., Solodov, M.V.: On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math. Program. 81, 23–35 (1998)

  4. 4.

    Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejer monotone methods in Hilbert space. Math. Oper. Res. 26, 248–264 (2001)

  5. 5.

    Bello, L., Raydan, M.: Preconditioned spectral projected-gradient method on convex sets. J. Comput. Math. 23, 225–232 (2005)

  6. 6.

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

  7. 7.

    Bertsekas, D.P.: On the Goldstein–Levitin–Polyak gradient projection method. IEEE Trans. Automat. Contr. AC-21(2), 174–184 (1976)

  8. 8.

    Byrne, C.L.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 18, 441–453 (2004)

  9. 9.

    Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM Publications, Philadelphia (1983)

  10. 10.

    Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62, 261–275 (1993)

  11. 11.

    Ekeland, I., Themam, R.: Convex analysis and variational problems. In: Classic in Applied Mathematics, p. 28. SIAM, Philadelphia (1999)

  12. 12.

    Ermoliev, Y.M.: Methods for solving nonlinear extremal problems. Cybernet. 2, 1–17 (1966)

  13. 13.

    Hager, W.W., Park, S.: The gradient projection method with exact line search. J. Glob. Optim. 30, 103–118 (2004)

  14. 14.

    Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)

  15. 15.

    Iusem, A.N.: On the convergence properties of the projected gradient method for convex optimization. Comput. Appl. Math. 22(1), 37–52 (2003)

  16. 16.

    Khobotov, E.N.: A modification of the extragradient method for the solution of variational inequalities and some optimization problems. Zh. Vychisl. Mat. Mat. Fiz. 27, 1462–1473 (1987)

  17. 17.

    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)

  18. 18.

    Marcotte, P.: Applications of Khobotov’s algorithm to variational and network equlibrium problems. Inform. Syst. Oper. Res. 29, 258–270 (1991)

  19. 19.

    Maingé, P.E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-point problems. Pacific J. Optim. 3(3), 529–538 (2007)

  20. 20.

    Moudafi, A.: Viscosity approximations methods for fixed point problems. J. Math. Anal. Appl. 241, 46–55 (2000)

  21. 21.

    Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mappings. SIAM J. Optim. 16(4), 1230–1241 (2006)

  22. 22.

    Solodov, M.V., Tseng, P.: Modified projection methods for monotone variational inequalities. SIAM J. Control Optim. 34(5), 1814–1834 (1996)

  23. 23.

    Solodov, M.V., Zavriev, S.K.: Error stability properties of generalized gradient-type algorithms. J. Optim. Theory Appl. 98, 663–680 (1998)

  24. 24.

    Solodov, M.V.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37(3), 756–776 (1999)

  25. 25.

    Xiu, N., Wang, C., Kong, L.: A note on the gradient projection method with exact stepsize rule. J. Comput. Math. 25(2), 221–230 (2007)

  26. 26.

    Yamada, I., Ogura, N.: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25(7–8), 619–655 (2004)

  27. 27.

    Zeng, L.C., Yao, J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan. J. Math. 10(5), 1293–1303 (2006)

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Correspondence to Paul-Emile Maingé.

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Maingé, P. Strong Convergence of Projected Subgradient Methods for Nonsmooth and Nonstrictly Convex Minimization. Set-Valued Anal 16, 899–912 (2008). https://doi.org/10.1007/s11228-008-0102-z

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  • Convex minimization
  • Projected gradient method
  • Nonsmooth optimization
  • Viscosity method

Mathematics Subject Classifications (2000)

  • 90C25
  • 90C30
  • 65C25