Extragradient algorithms extended to equilibrium problemsFootnote¶

This article is dedicated to the Memory of W. Oettli.

References

• Anh , PN , Muu , LD , Nguyen , VH and Strodiot , JJ . 2005 . “ On the contraction and nonexpansiveness properties of the marginal mapping in generalized variational inequalities involving co-coercive operators ” . In Generalized Convexity and Generalized Monotonicity and Applications , Edited by: Eberhard , A , Hadjisavvas , N and Luc , DT . 89 – 111 . New York : Springer . Chapter 5
   Google Scholar
• Anh , PN , Muu , LD , Nguyen , VH and Strodiot , JJ . 2005 . Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities . J. Optimiz. Theory App. , 124 : 285 – 306 .
   Web of Science ®Google Scholar
• Antipin , AS , Budak , BA and Vasil'ev , FP . 2002 . A regularized continuous extragradient method of the first order with variable metric for problems of equilibrium programming . Di.. Equat. , 38 : 1683 – 1693 .
   Web of Science ®Google Scholar
• Aubin , JP and Ekeland , I . 1984 . Applied Nonlinear Analysis , New York : Wiley .
   Google Scholar
• Berge , C . 1968 . Topological Spaces , New York : MacMillan .
   Google Scholar
• Blum , E and Oettli , W . 1994 . From optimization and variational inequality to equilibrium problems . The Mathematics Student , 63 : 127 – 149 .
   Google Scholar
• Cohen , G . 1980 . Auxiliary problem principle and decomposition of optimization problems . J. Optimiz. Theory App. , 32 : 277 – 305 .
   Web of Science ®Google Scholar
• Cohen , G . 1988 . Auxiliary principle extended to variational inequalities . J. Optimiz. Theory App. , 59 : 325 – 333 .
   Web of Science ®Google Scholar
• Facchinei , F and Pang , JS . 2003 . Finite-Dimensional Variational Inequalities and Complementary Problems , NewYork : Springer-Verlag .
   Google Scholar
• El Farouq , N . 2001 . Pseudomonotone variational inequalities: convergence of auxiliary problem method . J. Optimiz. Theory App. , 111 : 305 – 325 .
   Web of Science ®Google Scholar
• Hue , TT , Strodiot , JJ and Nguyen , VH . 2004 . Convergence of the approximate auxiliary problem method for solving generalized variational inequalities . J. Optimiz. Theory App. , 121 : 119 – 145 .
   Web of Science ®Google Scholar
• Iusem , AN , Kassay , G and Sosa , W . 2009 . On certain conditions for the existence of solutions of equilibrium problems . Math. Program., Ser. B. , 116 : 259 – 273 .
   Web of Science ®Google Scholar
• Konnov , IV . 2000 . Combined Relaxation Methods for Variational Inequalities , Berlin : Springer-Verlag .
   Google Scholar
• Konnov , IV . 2003 . Application of the proximal point method to nonmonotone equilibrium problems . J. Optimiz. Theory App. , 119 : 317 – 333 .
   Web of Science ®Google Scholar
• Korpelevich , GM . 1976 . Extragradient method for finding saddle points and other problems . Matecon , 12 : 747 – 756 .
   Google Scholar
• Martinet , B . 1970 . Régularisation d'inéquations variationelles par approximations successives . Rev. Fr. Automat. Infor. , 4 : 154 – 159 .
   Google Scholar
• Mastroeni , G . 2000 . On auxiliary principle for equilibrium problems . Publicatione del Dipartimento di Mathematica dell'Universita di Pisa , 3 : 1244 – 1258 .
   Google Scholar
• Mastroeni , G . 2004 . Gap function for equilibrium problems . J. Global. Optim. , 27 : 411 – 426 .
   Web of Science ®Google Scholar
• Moudafi , A . 2003 . Second order differential proximal methods for equilibrium problems . Journal of Inequalities in Pure and Applied Mathematics , 4 : 1 – 7 .
   Google Scholar
• Moudafi , A . 1999 . Proximal point algorithm extended to equilibrium problem . J. Nat. Geometry , 15 : 91 – 100 .
   Google Scholar
• Muu , LD . 1984 . Stability property of a class of variational inequalities . Optimization , 15 : 347 – 353 .
   Google Scholar
• Muu , LD and Oettli , W . 1992 . Convergence of an adaptive penalty scheme for finding constraint equilibria . Nonlinear Anal.-Theor. , 18 : 1159 – 1166 .
   Web of Science ®Google Scholar
• Nguyen , VH . 2002 . Lecture Notes on Equilibrium Problems , Nha Trang : CIUF-CUD Summer School on Optimization and Applied Mathematics .
   Google Scholar
• Noor , MA . 2003 . Extragradient methods for pseudomonotone variational inequalities . J. Optimiz. Theory App. , 117 : 475 – 488 .
   Web of Science ®Google Scholar
• Noor , MA . 2004 . Auxiliary principle technique for equilibrium problems . J. Optimiz. Theory App. , 122 : 371 – 386 .
   Web of Science ®Google Scholar
• Quy , NV , Nguyen , VH and Muu , LD . 2005 . On the Cournot–Nash oligopolistic market equilibrium problem with concave cost function . 2 Epreprint. Hanoi Institute of Mathematics
   Google Scholar
• Rockafellar , RT . 1970 . Convex Analysis , Princeton, NJ : Princeton University Press .
   Google Scholar
• Rockafellar , RT . 1976 . Monotone operators and the proximal point algorithm . SIAM J. Control Optim. , 14 : 877 – 898 .
   Web of Science ®Google Scholar
• Salmon , G , Nguyen , VH and Strodiot , JJ . 2000 . Coupling the auxiliary problem principle and the epiconvergence theory to solve generalized variational inequalities . J. Optimiz. Theory App. , 104 : 629 – 657 .
   Web of Science ®Google Scholar
• Tan , NX . 2004 . On the existence of solutions of quasi-variational inclusion problems . J. Optimiz. Theory App., , 123 : 619 – 638 .
   Web of Science ®Google Scholar
• Nguyen , TTV , Strodiot , JJ and Nguyen , VH . 2009 . A Bundle method for solving equilibrium problems . Math. Program., Ser. B. , 116 : 529 – 552 .
   Web of Science ®Google Scholar