American Institute of Mathematical Sciences

• Previous Article
Carbon spot prices in equilibrium frameworks associated with climate change
• JIMO Home
• This Issue
• Next Article
The evolution of rectangular bin packing problem — A review of research topics, applications, and cited papers
doi: 10.3934/jimo.2021195
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

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. 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.
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
 [1] Giuseppe Marino, Hong-Kun Xu. Convergence of generalized proximal point algorithms. Communications on Pure and Applied Analysis, 2004, 3 (4) : 791-808. doi: 10.3934/cpaa.2004.3.791 [2] Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations and Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017 [3] Ram U. Verma. On the generalized proximal point algorithm with applications to inclusion problems. Journal of Industrial and Management Optimization, 2009, 5 (2) : 381-390. doi: 10.3934/jimo.2009.5.381 [4] Guoyong Gu, Junfeng Yang. A unified and tight linear convergence analysis of the relaxed proximal point algorithm. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022107 [5] Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure and Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107 [6] Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013 [7] Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683 [8] András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks and Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43 [9] Samira Shahsavari, Saeed Ketabchi. The proximal methods for solving absolute value equation. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 449-460. doi: 10.3934/naco.2020037 [10] Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2417-2442. doi: 10.3934/dcdsb.2018259 [11] Yu-Lin Chang, Jein-Shan Chen, Jia Wu. Proximal point algorithm for nonlinear complementarity problem based on the generalized Fischer-Burmeister merit function. Journal of Industrial and Management Optimization, 2013, 9 (1) : 153-169. doi: 10.3934/jimo.2013.9.153 [12] Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107 [13] Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537 [14] Rafael de la Rosa, María Santos Bruzón. Differential invariants of a generalized variable-coefficient Gardner equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 747-757. doi: 10.3934/dcdss.2018047 [15] Hadi Khatibzadeh, Vahid Mohebbi, Mohammad Hossein Alizadeh. On the cyclic pseudomonotonicity and the proximal point algorithm. Numerical Algebra, Control and Optimization, 2018, 8 (4) : 441-449. doi: 10.3934/naco.2018027 [16] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [17] Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2213-2219. doi: 10.3934/cpaa.2012.11.2213 [18] Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations and Control Theory, 2021, 10 (2) : 249-258. doi: 10.3934/eect.2020064 [19] Igor Griva, Roman A. Polyak. Proximal point nonlinear rescaling method for convex optimization. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 283-299. doi: 10.3934/naco.2011.1.283 [20] Luca Lussardi, Stefano Marini, Marco Veneroni. Stochastic homogenization of maximal monotone relations and applications. Networks and Heterogeneous Media, 2018, 13 (1) : 27-45. doi: 10.3934/nhm.2018002

2020 Impact Factor: 1.801