An accelerated differential equation system for generalized equations

Qiyuan Wei; Liwei Zhang

doi:10.3934/jimo.2021195

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2023, Volume 19, Issue 1: 529-548. Doi: 10.3934/jimo.2021195

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An accelerated differential equation system for generalized equations

Qiyuan Wei^, and
Liwei Zhang

Institute of Operations Research and Control Theory, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

^* Corresponding author: Qiyuan Wei
^* Corresponding author: Qiyuan Wei

Received: May 2021

Early access: November 2021

Published: January 2023

Supported by the National Natural Science Foundation of China under project No.11971089 and No.11731013. Partially supported by Dalian High-level Talent Innovation Project No. 2020RD09.

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Abstract

An accelerated differential equation system with Yosida regularization and its numerical discretized scheme, for solving solutions to a generalized equation, are investigated. Given a maximal monotone operator $ T $ on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation

$ \begin{equation} \dot{x}(t)+T_{\lambda(t)}(x(t)-\alpha(t)T_{\lambda(t)}(x(t))) = 0,\quad t\geq t_0\geq 0, \end{equation} $

to the solution set $ T^{-1}(0) $ of a generalized equation $ 0 \in T(x) $. With smart choices of parameters $ \lambda(t) $ and $ \alpha(t) $, we prove the weak convergence of the trajectory to some point of $ T^{-1}(0) $ with $ \|\dot{x}(t)\|\leq {\rm O}(1/t) $ as $ t\rightarrow +\infty $. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory $ x^{h}(t) \rightarrow x(t) $ on interval $ [0,+\infty) $ is demonstrated when the step size $ h \rightarrow 0 $.

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Mathematics Subject Classification: 90C30.

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Table 1. Convergence of multipliers with different $ h $ step size

$ h $	$ \mu_1 $	$ \mu_2 $	$ \mu_3 $
0.5	0.4004	1.0000	1.1990
0.1	0.3714	1.0000	1.2570
0.05	0.3681	1.0000	1.2640
0.02	0.3661	1.0000	1.2580
0.01	0.3655	1.0000	1.2690
0.005	0.3651	1.0000	1.2700
0.001	0.3649	1.0000	1.2700

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References

[1]	A. S. Antipin, On differential prediction-type gradient methods for computing fixed points of extremal mappings, Differential Equations, 31 (1995), 1754-1763.
[2]	H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and Optimization, Second edition, Philadelphia, PA: MOS-SIAM Series on Optimization, 17, 2014. doi: 10.1137/1.9781611973488.
[3]	H. Attouch and J. Peypouquet, Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators, Mathematical Programming, 174 (2019), 391-432. doi: 10.1007/s10107-018-1252-x.
[4]	J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Berlin, Springer, 1984. doi: 10.1007/978-3-642-69512-4.
[5]	V. Barbu, A class of boundary value problems for second order abstract differential equations, J. Fat. Sci., Tokyo, Sect., 19 (1972), 295-319.
[6]	H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, 2011. doi: 10.1007/978-1-4419-9467-7.
[7]	H. Brezis, Equations d'evolution du second ordre associees a des operateurs monotones, Israel J. Math., 12 (1972), 51-60. doi: 10.1007/BF02764814.
[8]	R. E. Bruck, Asymptotic convergence of non-linear contraction semigroups in Hilbert space, J. of Functional Analysis, 18 (1975), 15-26. doi: 10.1016/0022-1236(75)90027-0.
[9]	A. Haraux, Sys'tems Dynamiques Dissipatifts et Applications, RMA 17, Masson, 1991.
[10]	F. J. Luque, Asymptotic convergence analysis of the proximal point algorithm, SIAM Journal on Control and Optimization, 22 (1984), 277-293. doi: 10.1137/0322019.
[11]	E. Mitidieri, Asymptotic behaviour of some second order evolution equations, Nonlinear Anal., 6 (1982), 1245-1252. doi: 10.1016/0362-546X(82)90033-5.
[12]	Y. Nesterov, A method of solving a convex programming problem with convergence rate O(1=k²), Soviet Mathmatics Doklady, 27 (1983), 372-376.
[13]	Z. Opial, Weak convegence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597. doi: 10.1090/S0002-9904-1967-11761-0.
[14]	S. M. Robinson, Generalized equations and their solutions, Part Ⅱ: Applications to nonlinear programming, Mathematical Programming Studies, 19 (1982), 200-221. doi: 10.1007/bfb0120989.
[15]	R. T. Rockafellar, Convex Analysis, Princeton, New Jersey: Princeton University Press, 1970.
[16]	R. T. Rockafellar, Monotone operators and the proximal point algotithm, SIAM Journal on Control and Optimization, 14 (1976), 877-898. doi: 10.1137/0314056.
[17]	R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 1 (1976), 97-116. doi: 10.1287/moor.1.2.97.
[18]	W. Su, S. Boyd and E. J. Candès, A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights, Neural Information Processing Systems, 27 (2014), 2510-2518.