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September  2018, 7(3): 373-401. doi: 10.3934/eect.2018019

Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems

Aicha Balhag , Zaki Chbani and  Hassan Riahi

Laboratory LIBMA Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 40000 Marrakech, Morroco

Received  February 2017 Revised  March 2018 Published  July 2018

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.

Keywords: Monotone bifunction, Cauchy problem, demipositive bifunction, asymptotic behaviour, convex minimisation, saddle point problem.
Mathematics Subject Classification: Primary: 37N40, 46N10, 49J40, 90C33.
Citation: Aicha Balhag, Zaki Chbani, Hassan Riahi. Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems. Evolution Equations and Control Theory, 2018, 7 (3) : 373-401. doi: 10.3934/eect.2018019
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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