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Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient
Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems
Laboratory LIBMA Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 40000 Marrakech, Morroco |
We consider the regularized Tikhonov-like dynamical equilibrium problem: find $u: [0, +∞ [\to\mathcal H$ such that for a.e. $t \ge 0$ and every $y∈K$, $\langle \dot{u}(t), y-u(t)\rangle +F(u(t), y)+\varepsilon(t) \langle u(t), y-u(t)\rangle \ge 0$, where $F:K×K \to \mathbb{R}$ is a monotone bifunction, $K$ is a closed convex set in Hilbert space $\mathcal H$ and the control function $\varepsilon(t)$ is assumed to tend to 0 as $t \to +∞$. We first establish that the corresponding Cauchy problem admits a unique absolutely continuous solution. Under the hypothesis that $\int_{0}^{+∞} \varepsilon (t) dt <∞$, we obtain weak ergodic convergence of $u(t)$ to $x∈K$ solution of the following equilibrium problem $F(x, y) \ge 0, \;\forall y∈K$. If in addition the bifunction is assumed demipositive, we show weak convergence of $u(t)$ to the same solution. By using a slow control $\int_{0}^{+∞} \varepsilon (t) dt = ∞$ and assuming that the bifunction $F$ is 3-monotone, we show that the term $\varepsilon (t)u(t)$ asymptotically acts as a Tikhonov regularization, which forces all the trajectories to converge strongly towards the element of minimal norm of the closed convex set of equilibrium points of $F$. Also, in the case where $\varepsilon $ has a slow control property and $\int_{0}^{+∞}\vert \dot{\varepsilon} (t) \vert dt < +∞ $, we show that the strong convergence property of $u(t)$ is satisfied. As applications, we propose a dynamical system to solve saddle-point problem and a neural dynamical model to handle a convex programming problem. In the last section, we propose two Tikhonov regularization methods for the proximal algorithm. We firstly use the prox-penalization algorithm $(ProxPA)$ by iteration $ x_{n+1} = J^{F_n}_{λ_n}(x_n)$ where $F_n(x, y) = F(x, y)+\varepsilon_n \langle x, y-x\rangle$, and $\varepsilon_n$ is the Liapunov parameter; afterwards, we propose the descent-proximal (forward-backward) algorithm $(DProxA)$: $x_{n+1} = J^F_{λ_n} ((1 - λ_n\varepsilon_n)x_n)$. We provide low conditions that guarantee a strong convergence of these algorithms to least norm element of the set of equilibrium points.
References:
[1] |
M. Ait Mansour, Z. Chbani and H. Riahi,
Recession bifunction and solvability of noncoercive equilibrium problems, Comm. Appl. Anal., 7 (2003), 369-377.
|
[2] |
M. H. Alizadeh, Monotone and Generalized Monotone Bifunctions and their Application to Operator Theory, Ph. D. Thesis, University of The Aegean, 2012. |
[3] |
H. Attouch and M. O. Czarnecki,
Asymptotic behavior of coupled dynamical systems with multiscale aspects, J. Differential Equations, 248 (2010), 1315-1344.
doi: 10.1016/j.jde.2009.06.014. |
[4] |
H. Attouch, A. Cabot and M. O. Czarnecki,
Asymptotic behavior of nonautonomous monotone and subgradient evolution equations, Trans. Amer. Math. Soc., 370 (2018), 755-790.
doi: 10.1090/tran/6965. |
[5] |
S. Bartz, H. H. Bauschke, J. Borwein, S. Reich and X. Wang,
Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative, Nonlinear Anal., 66 (2007), 1198-1223.
doi: 10.1016/j.na.2006.01.013. |
[6] |
E. Blum and W. Oettli,
From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.
|
[7] |
H. Brézis, Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution, Lecture Notes, vol. 5, North-Holland, 1972. |
[8] |
F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Appl. Math., vol. 18 (part 2), Amer. Math. Soc., Providence, RI, 1976, 1-308. |
[9] |
R. E. Bruck,
Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal., 18 (1975), 15-26.
doi: 10.1016/0022-1236(75)90027-0. |
[10] |
O. Chadli, Z. Chbani and H. Riahi,
Recession methods for equilibrium problems and applications to variational and hemivariational inequalities, Discrete Contin. Dyn. Syst., 5 (1999), 185-196.
|
[11] |
O. Chadli, Z. Chbani and H. Riahi,
Equilibrium problems and noncoercive variational inequalities, Optimization, 50 (2001), 17-27.
doi: 10.1080/02331930108844551. |
[12] |
O. Chadli, Z. Chbani and H. Riahi,
Equilibrium problems with generalized monotone bifunctions and Applications to Variational inequalities, J. Optim. Theory Appl., 105 (2000), 299-323.
doi: 10.1023/A:1004657817758. |
[13] |
Z. Chbani, Z. Mazgouri and H. Riahi, From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications, Accepted in Minimax Theory Appl. 04 (2019), No. 2. |
[14] |
Z. Chbani and H. Riahi,
Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166.
|
[15] |
Z. Chbani and H. Riahi,
Existence and asymptotic behaviour for solutions of dynamical equilibrium systems, Evol. Equ. Control Theory, 3 (2014), 1-14.
doi: 10.3934/eect.2014.3.1. |
[16] |
R. Cominetti, J. Peypouquet and S. Sorin,
Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization, J. Differential Equations, 245 (2008), 3753-3763.
doi: 10.1016/j.jde.2008.08.007. |
[17] |
X. P. Ding,
Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces, J. Optim. Theory Appl., 146 (2010), 347-357.
doi: 10.1007/s10957-010-9651-z. |
[18] |
B. V. Dinh and L. D. Muu, On penalty and gap function methods for bilevel equilibrium problems, J. Appl. Math., (2011), Art. ID 646452, 14 pp.
doi: 10.1155/2011/646452. |
[19] |
S. Efati and M. Baymani,
A new nonlinear neural network for solving convex nonlinear programming problems, Appl. Math. Comput., 168 (2005), 1370-1379.
doi: 10.1016/j.amc.2004.10.028. |
[20] |
K. Fan, A minimax inequality and application, in Inequalities, III (Proc. Third Sympos., UCLA, 1969. Dedicated to the Memory of T. S. Motgkin; O. Shisha, Ed. ), Academic Press, New York, (1972), 103-113. |
[21] |
N. Hadjisavvas and H. Khatibzadeh,
Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160.
doi: 10.1080/02331930801951116. |
[22] |
J. J. Hopfield and D. W. Tank,
Neural computation of decisions in optimization problems, Biol. Cybernet., 52 (1985), 141-152.
|
[23] |
P. G. Hung and L. D. Muu,
The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions, Nonlinear Anal., 74 (2011), 6121-6129.
doi: 10.1016/j.na.2011.05.091. |
[24] |
H. Khatibzadeh and S. Ranjbar,
On the strong convergence of halpern type proximal point algorithm, J. Optim. Theory Appl., 158 (2013), 385-396.
doi: 10.1007/s10957-012-0213-4. |
[25] |
N. Lehdili and A. Moudafi,
Combining the proximal algorithm and Tikhonov regularization, Optimization, 37 (1996), 239-252.
doi: 10.1080/02331939608844217. |
[26] |
F. Li,
Delayed Lagrangian neural networks for solving convex programming problems, Neural Comput., 73 (2010), 2266-2273.
doi: 10.1016/j.neucom.2010.01.009. |
[27] |
G. Mastroeni,
Gap functions for equilibrium problems, J. Global Optim., 27 (2003), 411-426.
doi: 10.1023/A:1026050425030. |
[28] |
A. Moudafi,
A recession notion for a class of monotone bivariate functions, Serdica Math. J., 26 (2000), 207-220.
|
[29] |
A. Moudafi,
Proximal methods for a class of bilevel monotone equilibrium problems, J. Global Optim., 47 (2010), 287-292.
doi: 10.1007/s10898-009-9476-1. |
[30] |
W. Oettli and M. Théra,
Equivalents of Ekeland's principle, Bull. Austral. Math. Soc., 48 (1993), 385-392.
doi: 10.1017/S0004972700015847. |
[31] |
Z. Opial,
Weak convergence 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. |
[32] |
G. B. Passty,
Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces, J. Math. Anal. Appl., 72 (1979), 383-390.
doi: 10.1016/0022-247X(79)90234-8. |
[33] |
J. Peypouquet, Analyse asymptotique de systèmes d'évolution et applications en optimisation, Ph. D. Thesis, UPMC Paris 6 and U. de Chile, 2007. |
[34] |
S. Reich,
Nonlinear evolution equations and nonlinear ergodic theorems, Nonlinear Anal., 1 (1976), 319-330.
doi: 10.1016/S0362-546X(97)90001-8. |
[35] |
A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, New York, 1977. |
[36] |
Y. Xia and J. Wang,
A recurrent neural network for solving nonlinear convex programs subject to linear constraints, IEEE Trans. Circuits Syst. I. Regul. Pap., 51 (2004), 1385-1394.
doi: 10.1109/TCSI.2004.830694. |
show all references
References:
[1] |
M. Ait Mansour, Z. Chbani and H. Riahi,
Recession bifunction and solvability of noncoercive equilibrium problems, Comm. Appl. Anal., 7 (2003), 369-377.
|
[2] |
M. H. Alizadeh, Monotone and Generalized Monotone Bifunctions and their Application to Operator Theory, Ph. D. Thesis, University of The Aegean, 2012. |
[3] |
H. Attouch and M. O. Czarnecki,
Asymptotic behavior of coupled dynamical systems with multiscale aspects, J. Differential Equations, 248 (2010), 1315-1344.
doi: 10.1016/j.jde.2009.06.014. |
[4] |
H. Attouch, A. Cabot and M. O. Czarnecki,
Asymptotic behavior of nonautonomous monotone and subgradient evolution equations, Trans. Amer. Math. Soc., 370 (2018), 755-790.
doi: 10.1090/tran/6965. |
[5] |
S. Bartz, H. H. Bauschke, J. Borwein, S. Reich and X. Wang,
Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative, Nonlinear Anal., 66 (2007), 1198-1223.
doi: 10.1016/j.na.2006.01.013. |
[6] |
E. Blum and W. Oettli,
From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.
|
[7] |
H. Brézis, Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution, Lecture Notes, vol. 5, North-Holland, 1972. |
[8] |
F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Appl. Math., vol. 18 (part 2), Amer. Math. Soc., Providence, RI, 1976, 1-308. |
[9] |
R. E. Bruck,
Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal., 18 (1975), 15-26.
doi: 10.1016/0022-1236(75)90027-0. |
[10] |
O. Chadli, Z. Chbani and H. Riahi,
Recession methods for equilibrium problems and applications to variational and hemivariational inequalities, Discrete Contin. Dyn. Syst., 5 (1999), 185-196.
|
[11] |
O. Chadli, Z. Chbani and H. Riahi,
Equilibrium problems and noncoercive variational inequalities, Optimization, 50 (2001), 17-27.
doi: 10.1080/02331930108844551. |
[12] |
O. Chadli, Z. Chbani and H. Riahi,
Equilibrium problems with generalized monotone bifunctions and Applications to Variational inequalities, J. Optim. Theory Appl., 105 (2000), 299-323.
doi: 10.1023/A:1004657817758. |
[13] |
Z. Chbani, Z. Mazgouri and H. Riahi, From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications, Accepted in Minimax Theory Appl. 04 (2019), No. 2. |
[14] |
Z. Chbani and H. Riahi,
Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166.
|
[15] |
Z. Chbani and H. Riahi,
Existence and asymptotic behaviour for solutions of dynamical equilibrium systems, Evol. Equ. Control Theory, 3 (2014), 1-14.
doi: 10.3934/eect.2014.3.1. |
[16] |
R. Cominetti, J. Peypouquet and S. Sorin,
Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization, J. Differential Equations, 245 (2008), 3753-3763.
doi: 10.1016/j.jde.2008.08.007. |
[17] |
X. P. Ding,
Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces, J. Optim. Theory Appl., 146 (2010), 347-357.
doi: 10.1007/s10957-010-9651-z. |
[18] |
B. V. Dinh and L. D. Muu, On penalty and gap function methods for bilevel equilibrium problems, J. Appl. Math., (2011), Art. ID 646452, 14 pp.
doi: 10.1155/2011/646452. |
[19] |
S. Efati and M. Baymani,
A new nonlinear neural network for solving convex nonlinear programming problems, Appl. Math. Comput., 168 (2005), 1370-1379.
doi: 10.1016/j.amc.2004.10.028. |
[20] |
K. Fan, A minimax inequality and application, in Inequalities, III (Proc. Third Sympos., UCLA, 1969. Dedicated to the Memory of T. S. Motgkin; O. Shisha, Ed. ), Academic Press, New York, (1972), 103-113. |
[21] |
N. Hadjisavvas and H. Khatibzadeh,
Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160.
doi: 10.1080/02331930801951116. |
[22] |
J. J. Hopfield and D. W. Tank,
Neural computation of decisions in optimization problems, Biol. Cybernet., 52 (1985), 141-152.
|
[23] |
P. G. Hung and L. D. Muu,
The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions, Nonlinear Anal., 74 (2011), 6121-6129.
doi: 10.1016/j.na.2011.05.091. |
[24] |
H. Khatibzadeh and S. Ranjbar,
On the strong convergence of halpern type proximal point algorithm, J. Optim. Theory Appl., 158 (2013), 385-396.
doi: 10.1007/s10957-012-0213-4. |
[25] |
N. Lehdili and A. Moudafi,
Combining the proximal algorithm and Tikhonov regularization, Optimization, 37 (1996), 239-252.
doi: 10.1080/02331939608844217. |
[26] |
F. Li,
Delayed Lagrangian neural networks for solving convex programming problems, Neural Comput., 73 (2010), 2266-2273.
doi: 10.1016/j.neucom.2010.01.009. |
[27] |
G. Mastroeni,
Gap functions for equilibrium problems, J. Global Optim., 27 (2003), 411-426.
doi: 10.1023/A:1026050425030. |
[28] |
A. Moudafi,
A recession notion for a class of monotone bivariate functions, Serdica Math. J., 26 (2000), 207-220.
|
[29] |
A. Moudafi,
Proximal methods for a class of bilevel monotone equilibrium problems, J. Global Optim., 47 (2010), 287-292.
doi: 10.1007/s10898-009-9476-1. |
[30] |
W. Oettli and M. Théra,
Equivalents of Ekeland's principle, Bull. Austral. Math. Soc., 48 (1993), 385-392.
doi: 10.1017/S0004972700015847. |
[31] |
Z. Opial,
Weak convergence 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. |
[32] |
G. B. Passty,
Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces, J. Math. Anal. Appl., 72 (1979), 383-390.
doi: 10.1016/0022-247X(79)90234-8. |
[33] |
J. Peypouquet, Analyse asymptotique de systèmes d'évolution et applications en optimisation, Ph. D. Thesis, UPMC Paris 6 and U. de Chile, 2007. |
[34] |
S. Reich,
Nonlinear evolution equations and nonlinear ergodic theorems, Nonlinear Anal., 1 (1976), 319-330.
doi: 10.1016/S0362-546X(97)90001-8. |
[35] |
A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, New York, 1977. |
[36] |
Y. Xia and J. Wang,
A recurrent neural network for solving nonlinear convex programs subject to linear constraints, IEEE Trans. Circuits Syst. I. Regul. Pap., 51 (2004), 1385-1394.
doi: 10.1109/TCSI.2004.830694. |
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