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Third Order Dynamical Systems for the Sum of Two Generalized Monotone Operators

Author

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  • Pham Viet Hai

    (Hanoi University of Science and Technology)

  • Phan Tu Vuong

    (University of Southampton)

Abstract

In this paper, we propose and analyze a third-order dynamical system for finding zeros of the sum of two generalized operators in a Hilbert space $$\mathcal {H}$$ H . We establish the existence and uniqueness of the trajectories generated by the system under appropriate continuity conditions, and prove exponential convergence to the unique zero when the sum of the operators is strongly monotone. Additionally, we derive an explicit discretization of the dynamical system, which results in a forward–backward algorithm with double inertial effects and larger range of stepsize. We establish the linear convergence of the iterates to the unique solution using this algorithm. Furthermore, we provide convergence analysis for the class of strongly pseudo-monotone variational inequalities. We illustrate the effectiveness of our approach by applying it to structured optimization and pseudo-convex optimization problems.

Suggested Citation

  • Pham Viet Hai & Phan Tu Vuong, 2024. "Third Order Dynamical Systems for the Sum of Two Generalized Monotone Operators," Journal of Optimization Theory and Applications, Springer, vol. 202(2), pages 519-553, August.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:2:d:10.1007_s10957-024-02437-y
    DOI: 10.1007/s10957-024-02437-y
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    References listed on IDEAS

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    1. Phan Tu Vuong & Jean Jacques Strodiot, 2020. "A Dynamical System for Strongly Pseudo-monotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 767-784, June.
    2. N. N. Tam & J. C. Yao & N. D. Yen, 2008. "Solution Methods for Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 253-273, August.
    3. M. Pappalardo & M. Passacantando, 2002. "Stability for Equilibrium Problems: From Variational Inequalities to Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 113(3), pages 567-582, June.
    4. E. Cavazzuti & M. Pappalardo & M. Passacantando, 2002. "Nash Equilibria, Variational Inequalities, and Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 114(3), pages 491-506, September.
    5. J. Bolte, 2003. "Continuous Gradient Projection Method in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 235-259, November.
    Full references (including those not matched with items on IDEAS)

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