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Relaxed-Inertial Proximal Point Algorithms for Nonconvex Equilibrium Problems with Applications

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Abstract

We propose a relaxed-inertial proximal point algorithm for solving equilibrium problems involving bifunctions which satisfy in the second variable a generalized convexity notion called strong quasiconvexity, introduced by Polyak (Sov Math Dokl 7:72–75, 1966). The method is suitable for solving mixed variational inequalities and inverse mixed variational inequalities involving strongly quasiconvex functions, as these can be written as special cases of equilibrium problems. Numerical experiments where the performance of the proposed algorithm outperforms one of the standard proximal point methods are provided, too.

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Data Availability

No data sets were generated during the current study. The used matlab codes are available from all authors on reasonable request.

References

  1. Addi, K., Adly, S., Goeleven, G., Saoud, H.: A sensitivity analysis of a class of semi-coercive variational inequalities using recession tools. J. Glob. Optim. 40, 7–27 (2008)

    Article  MathSciNet  Google Scholar 

  2. 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, 773–782 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Ansari, Q.H., Babu, F., Raju, M.S.: Proximal point method with Bregman distance for quasiconvex pseudomonotone equilibrium problems. Optimization (2023). https://doi.org/10.1080/02331934.2023.2252430

    Article  Google Scholar 

  5. Bauschke, H.H. , Combettes, P.L.: Convex Analysis and Monotone Operators Theory in Hilbert Spaces. CMS Books in Mathematics, 2nd edn. Springer, Berlin (2017)

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

    MathSciNet  Google Scholar 

  7. Boţ, R.I., Csetnek, E.R.: Proximal-gradient algorithms for fractional programming. Optimization 66, 1383–1396 (2017)

    Article  MathSciNet  PubMed  PubMed Central  Google Scholar 

  8. Brauer, A.: Limits for the characteristic roots of matrices IV: applications to stochastic matrices. Duke Math. J. 19, 75–91 (1952)

    Article  MathSciNet  Google Scholar 

  9. Bredies, K., Lorenz, D.: Iterated hard shrinkage for minimization problems with sparsity constraints. SIAM J. Sci. Comput. 30, 657–683 (2008)

    Article  MathSciNet  Google Scholar 

  10. Cambini, A., Martein, L.: Generalized Convexity and Optimization: Theory and Applications. Springer, Berlin (2009)

    Google Scholar 

  11. Cherukuri, A., Gharesifard, B., Cortés, J.: Saddle-point dynamics: conditions for asymptotic stability of saddle points. SIAM J. Control Optim. 55, 486–511 (2017)

    Article  MathSciNet  Google Scholar 

  12. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)

    Google Scholar 

  13. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. II. Springer, New York (2003)

    Google Scholar 

  14. Goeleven, D.: Complementarity and Variational Inequalities in Electronics. Academic Press, London (2017)

    Google Scholar 

  15. Goudou, X., Munier, J.: The gradient and heavy ball with friction dynamical systems: the quasiconvex case. Math. Program. 116, 173–191 (2007)

    Article  MathSciNet  Google Scholar 

  16. Grad, S.-M., Lara, F.: Solving mixed variational inequalities beyond convexity. J. Optim. Theory Appl. 190, 565–580 (2021)

    Article  MathSciNet  Google Scholar 

  17. Grad, S.-M., Lara, F., Marcavillaca, R.T.: Relaxed-inertial proximal point type algorithms for quasiconvex minimization. J. Glob. Optim. 85, 615–635 (2023)

    Article  MathSciNet  Google Scholar 

  18. Hadjisavvas, N.: Convexity, generalized convexity and applications. In: Al-Mezel, S., et al. (Eds.) Fixed Point Theory, Variational Analysis and Optimization, pp. 139–169. Taylor & Francis, Boca Raton (2014)

  19. Hadjisavvas, N., Komlosi, S., Schaible, S.: Handbook of Generalized Convexity and Generalized Monotonicity. Springer, Boston (2005)

    Book  Google Scholar 

  20. He, B.: Algorithm for a class of generalized linear variational inequality and its application. Sci. China A 25, 939–945 (1995)

    Google Scholar 

  21. He, B., He, X.-Z., Liu, H.K.: Solving a class of constrained black-box inverse variational inequalities. Eur. J. Oper. Res. 204, 391–401 (2010)

    Article  MathSciNet  Google Scholar 

  22. Hieu, D.: An inertial-like proximal algorithm for equilibrium problems. Math. Methods Oper. Res. 88, 399–415 (2018)

    Article  MathSciNet  Google Scholar 

  23. Hieu, D., Duong, H.N., Thai, B.H.: Convergence of relaxed inertial methods for equilibrium problems. J. Appl. Numer. Optim. 3, 215–229 (2021)

    Google Scholar 

  24. Iusem, A., Lara, F.: Optimality conditions for vector equilibrium problems with applications. J. Optim. Theory Appl. 180, 187–206 (2019)

    Article  MathSciNet  Google Scholar 

  25. Iusem, A., Lara, F.: Existence results for noncoercive mixed variational inequalities in finite dimensional spaces. J. Optim. Theory Appl. 183, 122–138 (2019)

    Article  MathSciNet  Google Scholar 

  26. Iusem, A., Lara, F.: A note on existence results for noncoercive mixed variational inequalities in finite dimensional spaces. J. Optim. Theory Appl. 187, 607–608 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Iusem, A., Lara, F., Marcavillaca , R.T., Yen, L.H. : A two-step PPA for nonconvex equilibrium problems with applications to fractional programming (submitted)(2023)

  29. Jovanović, M.: A note on strongly convex and quasiconvex functions. Math. Notes 60, 584–585 (1996)

    Article  MathSciNet  Google Scholar 

  30. Kassay, G., Rădulescu, V.: Equilibrium Problems and Applications. Elsevier, Amsterdam (2018)

    Google Scholar 

  31. Koumatos, K., Spirito, S.: Quasiconvex elastodynamics: weak–strong uniqueness for measure-valued solutions. Commun. Pure Appl. Math. 72, 1288–1320 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  33. Lara, F.: On nonconvex pseudomonotone equilibrium problems with applications. Set-Valued Var. Anal. 30, 355–372 (2022)

    Article  MathSciNet  Google Scholar 

  34. Maingué, M.: Asymptotic convergence of an inertial proximal method for unconstrained quasiconvex minimization. J. Glob. Optim. 45, 631–644 (2009)

    Article  MathSciNet  Google Scholar 

  35. Malitsky, Y.: Golden ratio algorithms for variational inequalities. Math. Program. 184, 383–410 (2020)

    Article  MathSciNet  Google Scholar 

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

  37. Ovcharova, N., Gwinner, J.: Semicoercive variational inequalities: from existence to numerical solutions of nonmonotone contact problems. J. Optim. Theory Appl. 171, 422–439 (2016)

    Article  MathSciNet  Google Scholar 

  38. Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966) (translation from Dokl Akad Nauk SSSR, 166, 287–290 (1966))

  39. Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. U.S.S.R. Comput. Math. Math. Phys. 4, 1–17 (1967) (translation from Zh. Vychisl. Mat. Mat. Fiz., 4, 791–803 (1964))

  40. Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–766 (2008)

    Article  MathSciNet  Google Scholar 

  41. Rockafellar, R.T., Wets, R.: Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time. SIAM J. Control Optim. 28, 810–820 (1990)

    Article  MathSciNet  Google Scholar 

  42. Rouhani, B.D. , Piranfar, M.R.: Asymptotic behavior for a quasi-autonomous gradient system of expansive type governed by a quasiconvex function. Electron. J. Differ. Equ. 2021, paper no. 15 (2021)

  43. Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, pp. 495–608. Kluwer, Dordrecht (1995)

    Chapter  Google Scholar 

  44. Schaible, S., Ziemba, W.T.: Generalized Concavity in Optimization and Economics. Academic Press, New York (1981)

    Google Scholar 

  45. Stancu-Minasian, I.M.: Fractional Programming: Theory, Methods and Applications. Kluwer, Dordrecht (1997)

    Book  Google Scholar 

  46. S̆tuller., J.: Ordered modified Gram–Schmidt orthogonalization revised. J. Comput. Appl. Math. 63, 221–227 (1995)

    Article  MathSciNet  Google Scholar 

  47. Van Vinh, L., Tran, V.N., Vuong, P.T.: A second-order dynamical system for equilibrium problems. Numer. Algorithms 91, 327–351 (2022)

    Article  MathSciNet  Google Scholar 

  48. Verkhovsky, B.S.: Information protection based on extraction of square roots of gaussian integers. Int. J. Commun. Netw. Syst. Sci. 4, 133–138 (2011)

    Google Scholar 

  49. Vial, J.P.: Strong convexity of sets and functions. J. Math. Econ. 9, 187–205 (1982)

    Article  MathSciNet  Google Scholar 

  50. Wang, M.: The existence results and Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. Ann. Math. Phys. 7, 151–163 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  51. Yen, L.H., Muu, L.D.: A parallel subgradient projection algorithm for quasiconvex equilibrium problems under the intersection of convex sets. Optimization 71, 4447–4462 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  53. Zhao, Y.-B.: The iterative methods for monotone generalized variational inequalities. Optimization 42, 285–307 (1997)

    Article  MathSciNet  Google Scholar 

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Funding

The authors would like to thank the MATH AmSud cooperation program (Project AMSUD-220020) for its support. This research was partially supported by Anid–Chile under project Fondecyt Regular 1220379 (Lara), and by a CIAS Senior Research Fellow Grant of the Corvinus Institute for Advanced Studies, by the Hi! PARIS Center and by a public grant from the Fondation Mathématique Jacques Hadamard as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH (Grad).

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Authors and Affiliations

  1. Unité de Mathématiques Appliquées, ENSTA Paris, Institut Polytechnique de Paris, 91120, Palaiseau, France

    Sorin-Mihai Grad

  2. Instituto de Alta investigación (IAI), Universidad de Tarapacá, Arica, Chile

    Felipe Lara & Raúl Tintaya Marcavillaca

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  1. Sorin-Mihai Grad
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  2. Felipe Lara
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  3. Raúl Tintaya Marcavillaca
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All authors contributed equally to the study conception, design and implementation and wrote and corrected the manuscript.

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Correspondence to Felipe Lara.

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Communicated by Alexander Vladimirovich Gasnikov.

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Grad, SM., Lara, F. & Tintaya Marcavillaca, R. Relaxed-Inertial Proximal Point Algorithms for Nonconvex Equilibrium Problems with Applications. J Optim Theory Appl (2024). https://doi.org/10.1007/s10957-023-02375-1

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