Proximal methods for a class of bilevel monotone equilibrium problems. (English) Zbl 1190.90125

Summary: We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a simple proximal method. Under mild conditions, the weak convergence of the sequences generated by the algorithm is obtained. Using this result we obtain corollaries which improve several corresponding results in this field.


90C25 Convex programming
Full Text: DOI


[1] Bauschke H.H., Combettes P.L.: A weak-to-strong convergence principle for Fejer monotone methods in Hilbert space. Math. Oper. Res. 26, 248–264 (2001) · Zbl 1082.65058
[2] Blum E., Oettli W.: From optimization and variational inequalities to equilibriums problems. Math. Stud. 63, 123–145 (1994) · Zbl 0888.49007
[3] Cabot A.: Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization. SIAM J. Optim. 15(2), 555–572 (2005) · Zbl 1079.90098
[4] Chadli O., Chbani Z., Riahi H.: Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities. J. Optim. Theory Appl. 105(2), 299–323 (2000) · Zbl 0966.91049
[5] Combettes P.L., Hirstoaga A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005) · Zbl 1109.90079
[6] Konnov I.V.: Application of the proximal method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003) · Zbl 1084.49009
[7] Luo Z.-Q., Pang J.-S., Ralph D.: Mathematical Programs With Equilibrium Constraints. Cambridge University Press, Cambridge (1996)
[8] Mastroeni G.: On auxiliary principle for equilibrium problems. Publicatione del Departimento di Mathematica dell’Universita di Pisa 3, 1244–1258 (2000)
[9] Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. (2006) · Zbl 1095.47038
[10] Moudafi A.: Proximal point algorithm extended for equilibrium problems. J. Nat. Geom. 15, 91–100 (1999) · Zbl 0974.65066
[11] Moudafi, A., Théra, M.: Proximal and dynamical approaches to equilibrium problems. Lecture Notes in Economics and Mathematical Systems, # 477, pp. 187–201, Springer, Berlin (1999) · Zbl 0944.65080
[12] Muu, L.D., Nguyen, V.H., Strodiot, J.J.: A linearly convergent algorithm for strongly monotone equilibrium problems: application to inexact prox-point methods (submited)
[13] Opial Z.: Weak convergence of the sequence of successive aapproximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967) · Zbl 0179.19902
[14] Solodov M.: An explicit descent method for bilevel convex optimization. J. Convex Anal. 14(2), 227–237 (2007) (to appear) · Zbl 1145.90081
[15] Solodov M., Svaiter B.F.: Error bounds for proximal point subproblems and associated inexat proximal point algorithm. Math. Prog. 88, 371–389 (2000) · Zbl 0963.90064
[16] Takahashi S., Takahashi W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331(1), 506–515 (2007) · Zbl 1122.47056
[17] Tran D.Q., Muu L.D., Nguyen V.H.: Extragradient algorithms extended to solving equilibrium problems. Optimization 57(6), 749–776 (2008) · Zbl 1152.90564
[18] Van N.T.T., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving equilibrium problems. Math. Program. 116(1–2), Ser. B, 529–552 (2009) · Zbl 1155.49006
[19] Yamada I., Ogura N.: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Num. Funct. Anal. Optim. 25(7–8), 619–655 (2004) · Zbl 1095.47049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.