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Linear conditioning, weak sharpness and finite convergence for equilibrium problems

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Abstract

The present paper first provides sufficient conditions and characterizations for linearly conditioned bifunction associated with an equilibrium problem. It then introduces the notion of weak sharp solution for equilibrium problems which is analogous to the linear conditioning notion. This new notion generalizes and unifies the notion of weak sharp minima for optimization problems as well as the notion of weak sharp solutions for variational inequality problems. Some sufficient conditions and characterizations of weak sharpness are also presented. Finally, we study the finite convergence property of sequences generated by some algorithms for solving equilibrium problems under linear conditioning and weak shapness assumptions. An upper bound of the number of iterations by which the sequence generated by proximal point algorithm converges to a solution of equilibrium problems is also given.

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References

  1. Alshahrani, M., Al-Homidan, S., Ansari, Q.H.: Weak sharp solutions for equilibrium problems in metric spaces. J. Nonlinear Convex Anal. 16(7), 1185–1193 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Al-Homidan, S., Ansari, Q.H., Burachik, R.: Weak sharp solutions for generalized variational inequalities. Positivity 21, 1067–1088 (2017)

    Article  MathSciNet  Google Scholar 

  3. Al-Homidan, S., Ansari, Q.H., Kassay, G.: Takahashi’s minimization theorem and some related results in quasi-metric spaces. J. Fixed Point Theory Appl. 21(1), Art. 38 (2019)

  4. Al-Homidan, S., Ansari, Q.H., Nguyen, L.V.: Weak sharp solutions for nonsmooth variational inequalities. J. Optim. Theory Appl. 175, 683–701 (2017)

    Article  MathSciNet  Google Scholar 

  5. Al-Homidan, S., Ansari, Q.H., Nguyen, L.V.: Finite convergence analysis and weak sharp solutions for variational inequalities. Optim. Lett. 11, 1647–1662 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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. Bot, R.I., Grad, S.-M.: Approaching the maximal monotonicity of bifunctions via representative functions. J. Convex Anal. 19, 713–724 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Burke, J.V., Deng, S.: Weak sharp minima revisited, part I: basic theory. Control Cybern. 31, 439–469 (2002)

    MATH  Google Scholar 

  10. Burke, J.V., Deng, S.: Weak sharp minima revisited, part II: application to linear regularity and error bounds. Math. Program. 104, 235–261 (2005)

    Article  MathSciNet  Google Scholar 

  11. Burke, J.V., Deng, S.: Weak sharp minima revisited, part III: error bounds for differentiable convex inclusions. Math. Program. 116, 37–56 (2009)

    Article  MathSciNet  Google Scholar 

  12. Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340–1359 (1993)

    Article  MathSciNet  Google Scholar 

  13. Burachik, R.S., Iusem, A.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer, New York (2008)

    MATH  Google Scholar 

  14. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) [Reprinted as Classics Appl. Math. 5, SIAM, Philadelphia (1990)]

  15. Combettes, P.L., Hirstoaga, A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Cromme, L.: Strong uniqueness. Numer. Math. 29, 179–193 (1978)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  18. Ferris, M.C.: Weak Sharp Minima and Penalty Functions in Mathematical Programming, Ph.D. Thesis, University of Cambridge (1988)

  19. Ferris, M.C.: Finite termination of the proximal point algorithm. Math. Program. 50, 359–366 (1991)

    Article  MathSciNet  Google Scholar 

  20. Flam, S.D., Antipin, A.S.: Equilibrium programming and proximal-like algorithms. Math. Program. 78, 29–41 (1997)

    Article  MathSciNet  Google Scholar 

  21. Hadjisavvas, N., Khatibzadeh, H.: Maximal monotonicity of bifunctions. Optimization 59, 147–160 (2010)

    Article  MathSciNet  Google Scholar 

  22. Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)

    Book  Google Scholar 

  23. Huang, H., He, M.: Weak sharp solutions of mixed variational inequalities in Banach spaces. Optim. Lett. 12, 287–299 (2018)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  25. Lemaire, B.: About the convergence of the proximal method. In: Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 382, pp. 39–51. Springer, Berlin (1992)

  26. Lemaire, B.: Bonne position conditionnement et bon comportement asymptotique. Séminaire d’Analyse Convexe 22, Exp. No. 5 (1992)

  27. Liu, Y.: Weakly sharp solutions and finite convergence of algorithms for a variational inequality problem. Optimization 67, 329–340 (2018)

    Article  MathSciNet  Google Scholar 

  28. Liu, Y., Wu, Z.: Characterization of weakly sharp solutions of a variational inequality by its primal gap function. Optim. Lett. 10(3), 563–576 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Marcotte, P., Zhu, D.L.: Weak sharp solutions of variational inequalities. SIAM J. Optim. 9(1), 179–189 (1998)

    Article  MathSciNet  Google Scholar 

  31. Matsushita, S., Xu, L.: On finite convergence of iterative methods for variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 161, 701–715 (2014)

    Article  MathSciNet  Google Scholar 

  32. Moudafi, A.: On finite and strong convergence of a proximal method for equilibrium problems. Numer. Funct. Anal. Optim. 28, 1347–1354 (2007)

    Article  MathSciNet  Google Scholar 

  33. Nikaido, H., Isoda, K.: Note on noncooperative convex games. Pac. J. Math. 5, 807–815 (1955)

    Article  Google Scholar 

  34. Nguyen, L.V., Qin, X.: Weak sharpness and finite convergence for mixed variational inequalities. J. Appl. Numer. Optim. 1, 77–90 (2019)

    Google Scholar 

  35. Patriksson, M.: A Unified Framework of Descent Algorithms for Nonlinear Programs and Variational Inequalities. Ph.D. Thesis, Department of Mathematics, Linkoping Institute of Technology, Linkoping, Sweden (1993)

  36. Polyak, B.T.: Sharp Minima, Institute of Control Sciences Lecture Notes, Moscow, USSR (1979). Presented at the IIASA Workshop on Generalized Lagrangians and Their Applications, IIASA, Laxenburg, Austria (1979)

  37. Wu, Z.: Characterizations of weakly sharp solutions for a variational inequality with a pseudomonotone mapping. Eur. J. Oper. Res. 265, 448–453 (2018)

    Article  MathSciNet  Google Scholar 

  38. Wu, Z.L., Wu, S.Y.: Weak sharp solutions of variational inequalities in Hilbert spaces. SIAM J. Optim. 14(4), 1011–1027 (2004)

    Article  MathSciNet  Google Scholar 

  39. Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process. J. Math. Anal. Appl. 178, 301–308 (1993)

    Article  MathSciNet  Google Scholar 

  40. Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co., Inc., River Edge (2002)

    Book  Google Scholar 

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Acknowledgements

In this research, this paper was supported by the National Natural Science Foundation of China under Grant No.11401152. The main part of the first author was done when he was a visitor at Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China. L.V.N was also supported by the Research Fund for International Young Scientists under Grant No. 1181101157. The research part of the second author was done during his visit to KFUPM, Dhahran, Saudi Arabia. He is grateful to KFUPM for providing excellent research facilities to carry his part of research work. Authors are also grateful to the referees for providing valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Natural Sciences, Hong Duc University, Thanh Hoa, Vietnam

    Luong Van Nguyen

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India

    Qamrul Hasan Ansari

  3. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    Qamrul Hasan Ansari

  4. Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang, China

    Xiaolong Qin

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  1. Luong Van Nguyen
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  2. Qamrul Hasan Ansari
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  3. Xiaolong Qin
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Correspondence to Xiaolong Qin.

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Van Nguyen, L., Ansari, Q.H. & Qin, X. Linear conditioning, weak sharpness and finite convergence for equilibrium problems. J Glob Optim 77, 405–424 (2020). https://doi.org/10.1007/s10898-019-00869-9

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