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An extragradient algorithm for quasiconvex equilibrium problems without monotonicity

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Abstract

We attempt to provide an algorithm for approximating a solution of the quasiconvex equilibrium problem that was proved to exist by K. Fan 1972. The proposed algorithm is an iterative procedure, where the search direction at each iteration is a normal-subgradient, while the step-size is updated avoiding Lipschitz-type conditions. The algorithm is convergent to a \(\rho \)- quasi-solution with any positive \(\rho \) if the bifunction f is semistrictly quasiconvex in its second variable, while it converges to the solution when f is strongly quasiconvex. Neither monotoniciy nor Lipschitz property is required. The main subprogram needed to solve at each iteration is a proximal regularized minimization problem whose objective function is the sum of a quasiconvex function and the one \(\Vert .\Vert ^2\). We also discuss several cases where this global optimization problem can be solved efficiently.

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References

  1. Nimanaa, N., Farajzadehb, A.P., Petrot, N.: Adaptive subgradient method for the split quasiconvex feasibility problems. Optimization 65, 1885–1898 (2016)

    Article  MathSciNet  Google Scholar 

  2. Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Nonlinear Programming Techniques for Equilibria. Springer (2019)

    Book  MATH  Google Scholar 

  3. Konnov, I.: Combined Relaxation Methods for Variational Inequalities. In: Lecture Notes in Economics and Mathematical Systems, vol. 435. Springer Verlag (2001)

    Google Scholar 

  4. Bigi, G., Passacantando, M.: Descent and penalization techniques for equilibrium problems with nonlinear constraints. J. Optim. Theory Appl. 164, 804–818 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ha, N.T.T., Thanh, T.T.H., Hai, N.N., Manh, H.D., Dinh, B.V.: A note on the combination of equilibrium problems. Math. Methods Oper. Res. 91, 311–323 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hieu, D.V., Muu, L.D., Strodiot, J.J.: Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems. J. Comput. Appl. Math. 376, 112–844 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. 74, 6121–6129 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mastroeni, G.: On auxiliary principle for equilibrium problems, in: Publicatione del Dipartimento di Mathematica dell, Vol. 3, Universita di Pisa, 1244–1258 (2000)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Sosa, P., Santos, S.: Scheimberg, an inexact subgradient algorithm for equilibrium problems. J. Comput. Appl. Math. 30, 91–107 (2011)

    Google Scholar 

  12. Strodiot, J.J., Vuong, P.T., Nguyen, T.T.V.: A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces. J. Global Optim. 64, 159–178 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  14. Aussel, D.: Adjusted sublevel sets, normal operator and quasiconvex programming. SIAM J. Optim. 16, 358–367 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Aussel, D., Dutta, J., Pandit, T.: About the links between equilibrium problems and variational inequalities. In: Neogy, S.K., Ravindra, B., Dubey, B.D. (eds.) Mathematical Programming and Game Theory, pp. 115–130. Springer (2018)

    Chapter  MATH  Google Scholar 

  16. Cruz Neto, J.X., Lopes, J.O., Soares, P.A., Jr.: A minimization algorithm for equilibrium problems with polyhedral constraints. Optimization 65(5), 1061–1068 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Iusem, A., Lara, F.: Proximal point algorithms for quasiconvex pseudomonotone equilibrium problems. J. Optim. Theory Appl. 193, 443–461 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yen, L.H., Muu, L.D.: A subgradient method for equilibrium problems involving quasiconvex bifunction. Oper. Res. Lett. 48(5), 579–583 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yen, L.H., Muu, L.D.: A parallel subgradient projection algorithm for quasiconvex equilibrium problems under the intersection of convex sets. Optimization (2021). https://doi.org/10.1080/02331934.2021.1946057

    Article  MATH  Google Scholar 

  20. Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-Łojasiewicz inequality. Math. Oper. Res. 35(2), 438–457 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grad, S.-M., Lara, F.: An extension of the proximal point algorithm beyond convexity. J. Global Optim. 82, 313–329 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mangasarian, O.: Nonlinear Programming. McGraw-Hill, Newyork (1969)

    MATH  Google Scholar 

  23. Lara, F.: On strongly quasiconvex functions: existence results and proximal point algorithms. J. Optim. Theory Appl. 192, 891–911 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  24. Greenberg, H.P., Pierskalla, W.P.: Quasi-conjugate functions and surogate duality. Cahiers Centre tudes Recherche Oper. 15, 437–448 (1973)

    MATH  Google Scholar 

  25. Penot, J.-P., Zalinescu, C.: Elements of quasiconvex subdifferential calculus. J. Convex Anal. 7, 243–269 (2000)

    MathSciNet  MATH  Google Scholar 

  26. Kiwiel, K.C.: Convergence and efficiency of subgradient methods for quasiconvex minimization. Math. Program. Ser. A 90, 1–25 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hu, Y., Yang, X., Sim, C.K.: Inexact subgradient methods for quasiconvex optimization problems. Eur. J. Oper. Res. 240, 315–327 (2015)

    Article  MATH  Google Scholar 

  28. Hu, Y., Yang, X., Yu, C.K.W.: Subgradient methods for saddle point problems of quasiconvex optimization. Pure Appl. Funct. Anal. 2, 83–97 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Hu, Y., Li, J., Yu, C.K.W.: Convergence rates of subgradient methods for quasiconvex optimization problems. Comput. Optim. Appl. 77, 183–212 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Penot, J.-P.: Are Generalized Derivatives useful for Generalized Convex Functions? In: Crouzeix, J.P., Martinez-Legaz, J.E., Volle, M. (eds.) Generalized Convexity, Generalized Monotonicity: Recent Results Nonconvex Optimization and Its Applications, vol. 27. Springer, Boston (1998)

    Google Scholar 

  31. Le, D.: Muu: stability property of a class of variational inequalities. Optimization 15, 347–351 (1984)

    MathSciNet  Google Scholar 

  32. Hare, W., Sagastizábal, C.: Computing proximal points of nonconvex functions. Math. Program. 116, 221–258 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kaplan, A., Tichatschke, R.: Proximal point methods and nonconvex optimization. J. Global Optim. 13, 389–406 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Boţ, R.I., Csetnek, E.R., Nguyen, D.-K.: A proximal minimization algorithm for structured nonconvex and nonsmooth problems. SIAM J. Optim. 29(2), 1300–1328 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15(6), 959–972 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rockafellar, T.R., Roger, J.B.: Variational Analysis. Springer (1998)

    Book  MATH  Google Scholar 

  37. Vial, J.-P.: Strongl and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  38. Golestani, M., Sadeghi, H., Tavan, Y.: Nonsmooth multiobjective problems and generalized vector variational inequalities using quasi-efficiency. J. Optim. Theory Appl. 179, 896–916 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous reviewers for their constructive comments and suggestions which help us to improve the quality of the paper. The research of the second author was supported by the Vietnam Academy of Science and Technology under Grant Number CTTH00.01/22-23.

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Authors and Affiliations

  1. TIMAS, Thang Long University and Institute of Mathematics, VAST, Hanoi, Vietnam

    Le Dung Muu

  2. Institute of Mathematics, VAST, Hanoi, Vietnam

    Le Hai Yen

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  1. Le Dung Muu
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Muu, L.D., Yen, L.H. An extragradient algorithm for quasiconvex equilibrium problems without monotonicity. J Glob Optim (2023). https://doi.org/10.1007/s10898-023-01291-y

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