Long-Time Behavior of a Gradient System Governed by a Quasiconvex Function
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Long-Time Behavior of a Gradient System Governed by a Quasiconvex Function Mohsen Rahimi Piranfar1
· Hadi Khatibzadeh2
Received: 23 January 2020 / Accepted: 6 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 Moro¸sanu (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 Mathematics Subject Classification 34G20 · 47J35 · 39A30
1 Introduction The study of dynamics and algorithms to find critical points of a function is of crucial importance in partial differential equations, optimization, equilibrium theory, economics and game theory. There is a vast literature on approximating minimizers
Communicated by Nikolai Pavlovich Osmolovskii.
B
Mohsen Rahimi Piranfar [email protected] Hadi Khatibzadeh [email protected]
1
Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran
2
University of Zanjan, Zanjan, Iran
123
Journal of Optimization Theory and Applications
of convex functions via the convergence analysis of the gradient flow of first- and second-order evolution equations governed by the subdifferential of a proper lower semicontinuous convex function. In a real Hilbert space setting, Barbu [1,2] established the existence of a unique solution to an incomplete Cauchy problem (ICP, for short) associated with a maximal monotone operator; see the problem ICP in the next section. In [3], Moro¸sanu studied the asymptotic behavior of such solutions and showed that, if the maximal monotone operator is the subdifferential of a proper lower semicontinuous convex function with a nonempty set of minimizers, then solutions converge weakly toward a minimizer as time goes to infinity; see also [4, p. 83]. For more investigations on the weak and strong convergence of the solutions in monotone case, the reader is referred to [5–7]. In [8], the second author showed that the rate of convergence of the values for ICP is better than the corresponding rate in the case of the first-order gradient equation (steepest descent equation). This shows an importance of studying the asymptotic behavior of solutions to ICP. For some applications of the asymptotic behavior of such a system in economics, s
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