An inertial type iterative method with Armijo linesearch for nonmonotone equilibrium problems

Abstract

This paper studies an iterative method with inertial term extrapolation step for solving an equilibrium problem of nonmonotone bifunctions in real Hilbert spaces. The inertia term extrapolation step is introduced to speed up the rate of convergence of the iteration process. We obtain convergence result under some continuity and convexity assumptions on the bifunction and the condition that the solution set of the associated Minty equilibrium problem is nonempty. Numerical comparisons of our proposed method with some other related method in the literature are given.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14(3), 773–782 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9(1–2), 3–11 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Attouch, H., Goudon, X., Redont, P.: The heavy ball with friction. I. The continuous dynamical system. Commun. Contemp. Math. 2(1), 1–34 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Attouch, H., Czarnecki, M.O.: Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Differ. Equ. 179(1), 278–310 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Attouch, H., Peypouquet, J., Redont, P.: A dynamical approach to an inertial forward–backward algorithm for convex minimization. SIAM J. Optim. 24(1), 232–256 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Beck, A., Teboulle, M.: A fast iterative shrinkage–thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Bigi, G., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227(1), 1–11 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Bigi, G., Passacantando, M.: Gap functions for quasi-equilibria. J. Global Optim. 66(4), 791–810 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63(1–4), 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas–Rachford splitting for monotone inclusion. Appl. Math. Comput. 256, 472–487 (2015)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Bot, R.I., Csetnek, E.R.: An inertial alternating direction method of multipliers. Minimax Theory Appl. 1(1), 29–49 (2016)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Bot, R.I., Csetnek, E.R.: An inertial forward–backward–forward primal–dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71(3), 519–540 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

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

    Google Scholar 

  14. 14.

    Castellani, M., Giuli, M.: Refinements of existence results for relaxed quasimonotone equilibrium problems. J. Global Optim. 57(4), 1213–1227 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Chen, C., Chan, R.H., Ma, S., Yang, J.: Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8(4), 2239–2267 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Contreras, J., Klusch, M., Krawczyk, J.B.: Numerical solution to Nash–Cournot equilibria in coupled constraint electricity markets. IEEE Trans. Power Syst. 19, 195–206 (2004)

    Article  Google Scholar 

  17. 17.

    Daniele, P., Giannessi, F., Maugeri, A. (eds.): Equilibrium Problems and Variational Models, Nonconvex Optimization and its Application, vol. 68. Kluwer, Norwell (2003)

    Google Scholar 

  18. 18.

    Dinh, B.V., Kim, D.S.: Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space. J. Comput. Appl. Math. 302, 106–117 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, NewYork (2003)

    Google Scholar 

  20. 20.

    Giannessi, F.: On Minty variational principle. In: Giannessi, F., Komlósi, S., Rapcsá, T. (eds.) New Trends in Mathematical Programming, pp. 93–99. Kluwer, Dordreccht (1998)

    Google Scholar 

  21. 21.

    Giannessi, F., Maugeri, A., Pardalos, P.M. (eds.): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer, Dordrecht (2001)

    Google Scholar 

  22. 22.

    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    Google Scholar 

  23. 23.

    He, Y.R.: A new double projection algorithm for variational inequalities. J. Comput. Appl. Math. 185(1), 166–173 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52(3), 301–316 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Iusem, A.N., Sosa, W.: On the proximal point method for equilibrium problem in Hilbert spaces. Optimization 59(8), 1259–1274 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Jaiboon, C., Kumam, P., Humphries, U.W.: Weak convergence theorem for an extragradient method for variational inequality, equilibrium and fixed point problems. Bull. Malays. Math. Sci. Soc. 32(2)(2), 173–185 (2009)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Jaiboon, C., Kumam, P.: A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Fixed Point Theory Appl., Art. ID 374815 (2009)

  28. 28.

    Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 21(4), 1319–1344 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)

    Google Scholar 

  30. 30.

    Korpelevich, G.M.: Extragradient method for finding saddle points and other problems. Ékonom. i Mat. Metody 12(4), 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Kumam, P., Katchang, P.: A viscosity of extragradient approximation method for finding equilibrium problems and fixed point problems for nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 3(4), 475–486 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Kumam, W., Kumam, P.: Hybrid iterative scheme by relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization. Nonlinear Anal. Hybrid Syst. 3(4), 640–656 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Lorenz, D.A., Pock, T.: An inertial forward–backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51(2), 311–325 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Maingé, P.E.: Convergence theorem for inertial KM-type algorithms. J. Comput. Appl. Math. 219(1), 223–236 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344(2), 876–887 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Mastroeni, G.: Gap functions for equilibrium problems. J. Global Optim. 27(4), 411–426 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., Giannessi, F., Maugeri, A. (eds.) Equilibrium Problems and Variational Models, pp. 289–298. Kluwer, Dordrecht (2003)

    Google Scholar 

  38. 38.

    Minty, G.J.: On the generalization of a direct method of calculus of variations. Bull. Am. Math. Soc. 73, 315–321 (1967)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Moudafi, A.: Proximal point algorithm extended to equilibrium problems. J. Nat. Geom. 15(1–2), 91–100 (1999)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Moudafi, A.: On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 359(2), 508–513 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Muu, L.D.: Stability property of a class of variational inequalities. Math. Operationsforsch. Stat. Ser. Optim 15(3), 347–353 (1984)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18(12), 1159–1166 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash–Cournot equilibrium model. J. Optim. Theory Appl. 142(1), 185–204 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Nguyen, V.H.: Lecture Notes on Equilibrium Problems. CIUF-CUD Summer School on Optimization and Applied Mathematics, Nha Trang (2002)

  45. 45.

    Noor, M.A.: Extragradient methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 117(3), 475–488 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Ochs, P., Brox, T., Pock, T.: iPiasco: inertial proximal algorithm for strongly convex optimization. J. Math. Imaging Vis. 53(2), 171–181 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 791–803 (1964)

    MathSciNet  Google Scholar 

  48. 48.

    Quoc, T.D., Muu, L.D.: Iterative methods for solving monotone equilibrium problems via dual gap functions. Comput. Optim. Appl. 51(2), 708–728 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms extended to equilibrium problems. J. Global Optim. 52(1), 139–159 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Quoc, T.D., Muu, L.D., Nguyen, H.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Riech, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Comtemp. Math. 568, 225–240 (2012)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Scheimberg, S., Santos, P.S.M.: A relaxed projection method for finite dimensional equilibrium problems. Optimization 60(8–9), 1193–1208 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Shehu, Y.: Iterative method for fixed point problem, variational inequality and generalized mixed equilibrium problems with applications. J. Global Optim. 52(1), 57–77 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Shehu, Y.: Strong convergence theorem for nonexpansive semigroups and systems of equilibrium problems. J. Global Optim. 56(4), 1675–1688 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Stos̆ić, M., Xavier, J., Dodig, M.: Projection on the intersection of convex sets. Linear Algebra Appl. 09, 191–205 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Global Optim. 56(2), 373–397 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Van, N.T.T., Strodiot, J.J., Nguyen, V.H.: The interior proximal extragradient method for solving equilibrium problems. J. Global Optim. 44(2), 175–192 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Van, N.T.T., Strodiot, J.J., Nguyen, V.H., Voung, P.T.: An extragradient-type method for solving nonmonotone quasi-equilibrium problems. Optimization 67(5), 651–664 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Voung, P.T., Strodiot, J.J., Nguyen, V.H.: Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J. Optim. Theory Appl. 155(2), 605–627 (2012)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Olaniyi. S. Iyiola.

Additional information

The work of this author is based on the research supported supported wholly by the National Research Foundation (NRF) of South Africa (Grant Numbers: 111992). Opinions expressed and conclusions arrived at, are those of the authors and are not necessarily to be attributed to the NRF. Y. Shehu: The research of this author is supported by the Alexander von Humboldt-Foundation.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Iyiola, O.S., Ogbuisi, F.U. & Shehu, Y. An inertial type iterative method with Armijo linesearch for nonmonotone equilibrium problems. Calcolo 55, 52 (2018). https://doi.org/10.1007/s10092-018-0295-y

Download citation

Keywords

  • Armijo linesearch
  • Nonmonotone equilibrium problems
  • Bifunctions
  • Inertia term

Mathematics Subject Classification

  • 47H04
  • 54H25
  • 47H10