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

Mohsen Rahimi Piranfar () and Hadi Khatibzadeh ()
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Mohsen Rahimi Piranfar: Institute for Advanced Studies in Basic Sciences (IASBS)
Hadi Khatibzadeh: University of Zanjan

Journal of Optimization Theory and Applications, 2021, vol. 188, issue 1, No 8, 169-191

Abstract: 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.

Keywords: Second-order evolution equation; Asymptotic behavior; Quasiconvex function; Minimization; 34G20; 47J35; 39A30 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-020-01784-w

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