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A Two-Step Proximal Point Algorithm for Nonconvex Equilibrium Problems with Applications to Fractional Programming

Author

Listed:
  • Alfredo Iusem

    (Fundação Getúlio Vargas (FGV))

  • Felipe Lara

    (Universidad de Tarapacá)

  • Raúl T. Marcavillaca

    (Universidad de Chile)

  • Le Hai Yen

    (Vietnam Academy of Sciences and Technology (VAST))

Abstract

We present a proximal point type algorithm tailored for tackling pseudomonotone equilibrium problems in a Hilbert space which are not necessarily convex in the second argument of the involved bifunction. Motivated by the extragradient algorithm, we propose a two-step method and we prove that the generated sequence converges strongly to a solution of the nonconvex equilibrium problem under mild assumptions and, also, we establish a linear convergent rate for the iterates. Furthermore, we identify a new class of functions that meet our assumptions, and we provide sufficient conditions for quadratic fractional functions to exhibit strong quasiconvexity. Finally, we perform numerical experiments comparing our algorithm against two alternative methods for classes of nonconvex mixed variational inequalities.

Suggested Citation

  • Alfredo Iusem & Felipe Lara & Raúl T. Marcavillaca & Le Hai Yen, 2024. "A Two-Step Proximal Point Algorithm for Nonconvex Equilibrium Problems with Applications to Fractional Programming," Journal of Global Optimization, Springer, vol. 90(3), pages 755-779, November.
    • Handle: RePEc:spr:jglopt:v:90:y:2024:i:3:d:10.1007_s10898-024-01419-8
    DOI: 10.1007/s10898-024-01419-8
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    References listed on IDEAS

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    1. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    2. Alfredo Iusem & Felipe Lara, 2019. "Optimality Conditions for Vector Equilibrium Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 187-206, January.
    3. Phan Tu Vuong & Jean Jacques Strodiot & Van Hien Nguyen, 2012. "Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 605-627, November.
    4. A. Iusem & F. Lara, 2022. "Proximal Point Algorithms for Quasiconvex Pseudomonotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 443-461, June.
    5. F. Lara, 2022. "On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 891-911, March.
    6. Alfredo Iusem & Felipe Lara, 2020. "A Note on “Existence Results for Noncoercive Mixed Variational Inequalities in Finite Dimensional Spaces”," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 607-608, November.
    7. Nina Ovcharova & Joachim Gwinner, 2016. "Semicoercive Variational Inequalities: From Existence to Numerical Solution of Nonmonotone Contact Problems," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 422-439, November.
    8. Alfredo Iusem & Felipe Lara, 2019. "Existence Results for Noncoercive Mixed Variational Inequalities in Finite Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 122-138, October.
    9. Sorin-Mihai Grad & Felipe Lara, 2021. "Solving Mixed Variational Inequalities Beyond Convexity," Journal of Optimization Theory and Applications, Springer, vol. 190(2), pages 565-580, August.
    10. I.V. Konnov, 2003. "Application of the Proximal Point Method to Nonmonotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 317-333, November.
    11. A. Kabgani & F. Lara, 2022. "Strong subdifferentials: theory and applications in nonconvex optimization," Journal of Global Optimization, Springer, vol. 84(2), pages 349-368, October.
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