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

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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

Zaki Chbani ^1, and Hassan Riahi ^1,

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

Received April 2012 Revised November 2012 Published April 2013

Abstract
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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.

Keywords: penalty proximal algorithm, splitting algorithm, Bilevel equilibrium problem, fixed point, weak and strong convergence..

Mathematics Subject Classification: Primary: 46N10, 65K15, 90C33; Secondary: 47H05, 65K1.

Citation: Zaki Chbani, Hassan Riahi. Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems. Numerical Algebra, Control and 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.
[2]	J. P. Aubin, "Optima and Equilibria: An Introduction to Nonlinear Analysis," Springer, 2nd edition, 2002.
[3]	E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.
[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.
[5]	Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166.
[6]	N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160. doi: 10.1080/02331930801951116.
[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.
[8]	A. Moudafi, Proximal point algorithm extended for equilibrium problems, J. Nat. Geom., 15 (1999), 91-100.
[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.
[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.
[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.
[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.

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.
[2]	J. P. Aubin, "Optima and Equilibria: An Introduction to Nonlinear Analysis," Springer, 2nd edition, 2002.
[3]	E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.
[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.
[5]	Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166.
[6]	N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160. doi: 10.1080/02331930801951116.
[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.
[8]	A. Moudafi, Proximal point algorithm extended for equilibrium problems, J. Nat. Geom., 15 (1999), 91-100.
[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.
[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.
[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.
[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.

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