# American Institute of Mathematical Sciences

• Previous Article
Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming
• NACO Home
• This Issue
• Next Article
Solutions of the Yang-Baxter matrix equation for an idempotent
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.  doi: 10.1137/100789464.  Google Scholar [2] J. P. Aubin, "Optima and Equilibria: An Introduction to Nonlinear Analysis,", Springer, (2002).   Google Scholar [3] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems,, Math. Student, 63 (1994), 123.   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.  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.   Google Scholar [6] N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions,, Optimization, 59 (2010), 147.  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.   Google Scholar [8] A. Moudafi, Proximal point algorithm extended for equilibrium problems,, J. Nat. Geom., 15 (1999), 91.   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.  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.  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.  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.  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.  doi: 10.1137/100789464.  Google Scholar [2] J. P. Aubin, "Optima and Equilibria: An Introduction to Nonlinear Analysis,", Springer, (2002).   Google Scholar [3] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems,, Math. Student, 63 (1994), 123.   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.  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.   Google Scholar [6] N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions,, Optimization, 59 (2010), 147.  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.   Google Scholar [8] A. Moudafi, Proximal point algorithm extended for equilibrium problems,, J. Nat. Geom., 15 (1999), 91.   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.  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.  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.  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.  doi: 10.1016/0022-247X(79)90234-8.  Google Scholar
 [1] Hadi Khatibzadeh, Vahid Mohebbi, Mohammad Hossein Alizadeh. On the cyclic pseudomonotonicity and the proximal point algorithm. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 441-449. doi: 10.3934/naco.2018027 [2] Yibing Lv, Tiesong Hu, Jianlin Jiang. Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1743-1755. doi: 10.3934/dcdss.2020102 [3] Ram U. Verma. On the generalized proximal point algorithm with applications to inclusion problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 381-390. doi: 10.3934/jimo.2009.5.381 [4] 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 & Management Optimization, 2013, 9 (1) : 153-169. doi: 10.3934/jimo.2013.9.153 [5] Yue Zheng, Zhongping Wan, Shihui Jia, Guangmin Wang. A new method for strong-weak linear bilevel programming problem. Journal of Industrial & Management Optimization, 2015, 11 (2) : 529-547. doi: 10.3934/jimo.2015.11.529 [6] Xueling Zhou, Meixia Li, Haitao Che. Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020082 [7] Giuseppe Marino, Hong-Kun Xu. Convergence of generalized proximal point algorithms. Communications on Pure & Applied Analysis, 2004, 3 (4) : 791-808. doi: 10.3934/cpaa.2004.3.791 [8] Zheng-Hai Huang, Shang-Wen Xu. Convergence properties of a non-interior-point smoothing algorithm for the P*NCP. Journal of Industrial & Management Optimization, 2007, 3 (3) : 569-584. doi: 10.3934/jimo.2007.3.569 [9] Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092 [10] Jie Shen, Jian Lv, Fang-Fang Guo, Ya-Li Gao, Rui Zhao. A new proximal chebychev center cutting plane algorithm for nonsmooth optimization and its convergence. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1143-1155. doi: 10.3934/jimo.2018003 [11] Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020063 [12] Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial & Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333 [13] Qin Sheng, David A. Voss, Q. M. Khaliq. An adaptive splitting algorithm for the sine-Gordon equation. Conference Publications, 2005, 2005 (Special) : 792-797. doi: 10.3934/proc.2005.2005.792 [14] Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial & Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177 [15] Sanming Liu, Zhijie Wang, Chongyang Liu. Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 79-89. doi: 10.3934/naco.2015.5.79 [16] Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020066 [17] Zhiqing Meng, Qiying Hu, Chuangyin Dang. A penalty function algorithm with objective parameters for nonlinear mathematical programming. Journal of Industrial & Management Optimization, 2009, 5 (3) : 585-601. doi: 10.3934/jimo.2009.5.585 [18] Pilar Bayer, Dionís Remón. A reduction point algorithm for cocompact Fuchsian groups and applications. Advances in Mathematics of Communications, 2014, 8 (2) : 223-239. doi: 10.3934/amc.2014.8.223 [19] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [20] Ali Fuat Alkaya, Dindar Oz. An optimal algorithm for the obstacle neutralization problem. Journal of Industrial & Management Optimization, 2017, 13 (2) : 835-856. doi: 10.3934/jimo.2016049

Impact Factor:

## Metrics

• PDF downloads (46)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]