January  2023, 19(1): 529-548. doi: 10.3934/jimo.2021195

An accelerated differential equation system for generalized equations

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

* Corresponding author: Qiyuan Wei

Received  May 2021 Published  January 2023 Early access  November 2021

Fund Project: 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

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 $
.
Citation: Qiyuan Wei, Liwei Zhang. An accelerated differential equation system for generalized equations. Journal of Industrial and Management Optimization, 2023, 19 (1) : 529-548. doi: 10.3934/jimo.2021195
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. SuS. 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. 

show all references

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. SuS. 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. 

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
$ 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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