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Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities

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Abstract

In this paper we present the relaxed inertial proximal algorithm for Ky Fan minimax inequalities. Based on Opial lemma, we propose a weak convergence result to a solution of the problem by eliminating in the algorithm (RIPAFAN) the Browder–Halpern’s factor of contraction. We present after, a first result of strong convergence by adding a strong monotonicity condition. Secondly, we eliminate the strong monotonicity and add a Browder–Halpern’s contraction factor in the algorithm (RIPAFAN) and then ensure the strong convergence to a selected solution with respect to the contraction factor. Some examples are proposed. The first one concerns the convex minimization where the objective function is only controlled with a provided well conditioning. In the second one, we propose monotone set-valued variational inequalities. The last example deals with the problem of fixed point for a nonexpansive set-valued operator.

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References

  1. Alvarez F.: On the minimizing property of a second order dissipative dynamical system in Hilbert spaces. SIAM J. Control Optim. 39, 1102–1119 (2000)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Antipin A.S.: Second order controlled differential grradient methods for equilibrium problems. Differ. Equ. 35(5), 592–601 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Antipin A.S.: Equilibrium programming: proximal methods. Comput. Math. Phys. 37(11), 1285–1296 (1997)

    MathSciNet  Google Scholar 

  5. Antipin A.S., Flam S.: Equilibrium programming using proximal-like algorithms. Math. Program. 78(1), 29–41 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Brézis H., Nirenberg L., Stampacchia G.: A Remark on Ky Fan’s minimax principle. Bollettino Unione Matematica Italiana (III) VI, 129–132 (1972)

    Google Scholar 

  7. Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Browder F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear maps in Banach spaces. Arch. Rat. Mech. Anal. 24, 82–90 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chadli O., Chbani Z., Riahi H.: Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities. J. Optim. Theory Appl. 105, 299–323 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chbani Z., Riahi H.: Variational principle for monotone and maximal bifunctions. Serdica Math. J. 29, 159–166 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Fan K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972)

    Google Scholar 

  12. Giannessi, F., Maugeri, A., Pardalos Panos, M. (eds.): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer Academic Publishers, London (2002)

    Google Scholar 

  13. Göpfert A., Riahi H., Tammer Ch., Zalinescu C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  15. Martinet B.: Régularisation d’inéquations variationnelles par approximations successives, Revue Française d’Automatique et d’Informatique. Recherche Opérationnelle 4, 154–159 (1970)

    MathSciNet  MATH  Google Scholar 

  16. Mosco, U.: Implicit variational problems and quasivariational inequalities. Lecture Notes in Mathematics, vol. 543. Springer, Berlin, 83–156 (1976)

  17. Moudafi A.: Proximal point algorithm extended to equilibrium problems. J. Nat. Geom. 15, 91–100 (1999)

    MathSciNet  MATH  Google Scholar 

  18. Moudafi, A.: Second-order differential proximal methods for equilibrium problems. J. Inequal. Pure and Appl. Math. 4(1) Art. 18 (2003)

    Google Scholar 

  19. Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Suzuki T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xu H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116(3), 659–678 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Xu H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yuan G.X.Z.: KKM Theory and Applications in Nonlinear Analysis. Marcel Dekker Inc, New York (1999)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Sciences Semlalia, Mathematics, Cadi Ayyad University, 40000, Marrakech, Morroco

    Zaki Chbani & Hassan Riahi

Authors
  1. Zaki Chbani
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  2. Hassan Riahi
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Correspondence to Hassan Riahi.

Additional information

This paper is dedicated to the memory of Professors Ky Fan (1914–2010) and Werner Oettli (1933–1999).

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Chbani, Z., Riahi, H. Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities. Optim Lett 7, 185–206 (2013). https://doi.org/10.1007/s11590-011-0407-y

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  • DOI: https://doi.org/10.1007/s11590-011-0407-y

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