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2013, 3(2): 353-366. doi: 10.3934/naco.2013.3.353

Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems

1. 

Cadi Ayyad University, Faculty of Sciences Semlalia, Mathematics, 40000 Marrakech, Morocco, Morocco

Received  April 2012 Revised  November 2012 Published  April 2013

The aim of this paper is to obtain, in a Hilbert space $H$, the weak and strong convergence of a penalty proximal algorithm and a splitting one for a bilevel equilibrium problem: find $ x\in S_F $ such that $\ G(x,y)\geq 0\ $ for all $ \ y\in S_F$, where $S_F :=\lbrace y\in K\; :\; F(y,u)\geq 0\;\; \forall u\in K \rbrace$, and $F,G:K\times K\longrightarrow \mathbb{R}$ are two bifunctions with $K$ a nonempty closed convex subset of $H$. In our framework, results of convergence generalize those recently obtained by Attouch et al. (SIAM Journal on Optimization 21, 149-173 (2011)). We show in particular that for the strong convergence of the penalty algorithm, the geometrical condition they impose is not required. We also give applications of the iterative schemes to fixed point problems and variational inequalities.
Citation: Zaki Chbani, Hassan Riahi. Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 353-366. doi: 10.3934/naco.2013.3.353
References:
[1]

H. Attouch, M. O. Czarnecki and J. Peypouquet, Prox-penalization and splitting methods for constrained variational problems, SIAM J. Control Optim., 21 (2011), 149-173. doi: 10.1137/100789464.  Google Scholar

[2]

J. P. Aubin, "Optima and Equilibria: An Introduction to Nonlinear Analysis," Springer, 2nd edition, 2002. Google Scholar

[3]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.  Google Scholar

[4]

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.  Google Scholar

[5]

Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166.  Google Scholar

[6]

N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160. doi: 10.1080/02331930801951116.  Google Scholar

[7]

P. E. Mainge and A. Moudafi, Strong convergence of an iterative method for hierarchical fixed-points problems, Pacific J. Optim., 3 (2007), 529-538.  Google Scholar

[8]

A. Moudafi, Proximal point algorithm extended for equilibrium problems, J. Nat. Geom., 15 (1999), 91-100.  Google Scholar

[9]

A. Moudafi, On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces, J. Math Anal. Appl., 359 (2009), 508-513. doi: 10.1016/j.jmaa.2009.06.005.  Google Scholar

[10]

A. Moudafi, Proximal methods for a class of bilevel monotone equilibrium problems, J. Global Optimization, 47 (2010), 287-292. doi: 10.1007/s10898-009-9476-1.  Google Scholar

[11]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Aust. Math. Soc., 73 (1967), 591-597. doi: 10.1090/S0002-9904-1967-11761-0.  Google Scholar

[12]

G. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72 (1979), 383-390. doi: 10.1016/0022-247X(79)90234-8.  Google Scholar

show all references

References:
[1]

H. Attouch, M. O. Czarnecki and J. Peypouquet, Prox-penalization and splitting methods for constrained variational problems, SIAM J. Control Optim., 21 (2011), 149-173. doi: 10.1137/100789464.  Google Scholar

[2]

J. P. Aubin, "Optima and Equilibria: An Introduction to Nonlinear Analysis," Springer, 2nd edition, 2002. Google Scholar

[3]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.  Google Scholar

[4]

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.  Google Scholar

[5]

Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166.  Google Scholar

[6]

N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160. doi: 10.1080/02331930801951116.  Google Scholar

[7]

P. E. Mainge and A. Moudafi, Strong convergence of an iterative method for hierarchical fixed-points problems, Pacific J. Optim., 3 (2007), 529-538.  Google Scholar

[8]

A. Moudafi, Proximal point algorithm extended for equilibrium problems, J. Nat. Geom., 15 (1999), 91-100.  Google Scholar

[9]

A. Moudafi, On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces, J. Math Anal. Appl., 359 (2009), 508-513. doi: 10.1016/j.jmaa.2009.06.005.  Google Scholar

[10]

A. Moudafi, Proximal methods for a class of bilevel monotone equilibrium problems, J. Global Optimization, 47 (2010), 287-292. doi: 10.1007/s10898-009-9476-1.  Google Scholar

[11]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Aust. Math. Soc., 73 (1967), 591-597. doi: 10.1090/S0002-9904-1967-11761-0.  Google Scholar

[12]

G. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72 (1979), 383-390. doi: 10.1016/0022-247X(79)90234-8.  Google Scholar

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