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Long-Time Behavior of a Gradient System Governed by a Quasiconvex Function

Author

Listed:
  • Mohsen Rahimi Piranfar

    (Institute for Advanced Studies in Basic Sciences (IASBS))

  • Hadi Khatibzadeh

    (University of Zanjan)

Abstract

We consider a second-order differential equation governed by a quasiconvex function with a nonempty set of minimizers. Assuming that the gradient of this function is Lipschitz continuous, the existence of solutions to the gradient system is guaranteed. We study the asymptotic behavior of these solutions in continuous and discrete times. More precisely, we show that, if a solution is bounded, then it converges weakly to a critical point of the function; otherwise, it goes to infinity (in norm). We also provide several sufficient conditions for obtaining strong convergence in both continuous and discrete cases. Our work is motivated by an open problem proposed by Khatibzadeh and Moroşanu (J Convex Anal 26:1175–1186, 2019), and we solve this problem in the case, where the gradient of the function is Lipschitz continuous on bounded sets.

Suggested Citation

  • Mohsen Rahimi Piranfar & Hadi Khatibzadeh, 2021. "Long-Time Behavior of a Gradient System Governed by a Quasiconvex Function," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 169-191, January.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:1:d:10.1007_s10957-020-01784-w
    DOI: 10.1007/s10957-020-01784-w
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    References listed on IDEAS

    as
    1. Brock, William A. & Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2014. "Spatial externalities and agglomeration in a competitive industry," Journal of Economic Dynamics and Control, Elsevier, vol. 42(C), pages 143-174.
    2. Brock, William A. & Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2014. "Optimal agglomerations in dynamic economics," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 1-15.
    3. Hadi Khatibzadeh, 2012. "On the Rate of Convergence of Gradient Flow for Some Evolution Systems," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 685-690, August.
    Full references (including those not matched with items on IDEAS)

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