Abstract
We consider a generalized equilibrium problem involving DC functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for DC programming problems. The first one allows us to obtain a unified dual analysis for many interesting problems. So, this dual coincides with the dual problem proposed by Martinez-Legaz and Sosa (J Glob Optim 25:311–319, 2006) for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco’s dual problem (Mosco in J Math Anal Appl 40:202–206, 1972) when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by Jacinto and Scheimberg (Optimization 57:795–805, 2008) for convex equilibrium problems. Through these schemes, as by products, we obtain new optimality conditions for (GEP) and also, gap functions for (GEP), which cover the ones in Antangerel et al. (J Oper Res 24:353–371, 2007, Pac J Optim 2:667–678, 2006) for variational inequalities and standard convex equilibrium problems. These results, in turn, when applied to DC and convex optimization problems with convex constraints (considered as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for DC problems in Dinh et al. (Optimization 1–20, 2008, J Convex Anal 15:235–262, 2008), Fenchel-Lagrange and Lagrange dualities for convex problems as in Antangerel et al. (Pac J Optim 2:667–678, 2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al. (Applied Mathematics research report AMR04/8, 2004). Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Antangerel L., Bot R.I., Wanka G.: On the construction of gap functions for variational inequalities via conjugate duality. Asia-Pac. J. Oper. Res. 24, 353–371 (2007)
Antangerel L., Bot R.I., Wanka G.: On gap functions for equilibrium problems via Fenchel duality. Pac. J. Optim. 2, 667–678 (2006)
Attouch H., Brezis H.: Duality for the sum of convex functions in general Banach spaces. In: Barroso, J.A. (eds) Aspects of Mathematics and its Application, pp. 125–133. Elsevier, Amsterdam, The Netherlands (1986)
Auslender A.: Optimisation. Méthodes Numériques. Masson, Paris (1976)
Bigi G., Castellani M., Kassay G.: A dual view of equilibrium problems. J. Math. Anal. Appl. 342, 17–26 (2008)
Blum E., Oettli W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 127–149 (1994)
Bot, R.I., Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Anal, to appear.
Burachik R.S., Jeyakumar V.: A new geometric condition for Fenchel’s duality in infinite dimensional spaces. Math. Program. 104(B), 229–233 (2005)
Burachik R.S., Jeyakumar V.: A dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12, 279–290 (2005)
Dinh N., Goberna M.A., López M.A., Son T.Q.: New Farkas-type results with applications to convex infinite programming. ESAIM: Control Optim. Cal. Var. 13, 580–597 (2007)
Dinh N., Jeyakumar V., Lee G.M.: Sequential Lagrangian conditions for convex programs with applications to semi-definite programming. J. Optim. Theory Appl. 125, 85–112 (2005)
Dinh, N., Mordukhovich, B.S., Nghia, T.T.A.: Subdifferentials of value functions and optimality conditions for some classes of DC and bilevel infinite and semi-infinite programs. Research Report # 4, Department of Mathematics, Wayne State University, Detroit, Michigan (2008) (to appear in Math. Program.)
Dinh, N., Nghia, T.T.A., Vallet, G.: A closedness condition and its applications to DC programs with convex constraints. Optimization, 1-20, iFirst (2008) doi:10.1080/02331930801951348 First Published on: 31 March 2008 http://www.informaworld.com/smpp/title~content=g770174694~db=all?stem=3#messages
Dinh N., Vallet G., Nghia T.T.A.: Farkas-type results and duality for DC programs with convex constraints. J. Convex Anal. 15, 235–262 (2008)
Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for extended Farkas’ lemmas and Lagrangian dualities in convex infinite programming (Submitted).
Fukushima, M.: A class of gap functions for quasi-variational inequality problems (Preprint)
Hiriart-Urruty J.B.: From convex optimization to non-convex optimization necessary and sufficient conditions for global optimality. In: Gilbert, R.P., Panagiotopoulos, P.D., Pardalos, P.M. (eds) From Convexity to Non-convexity, pp. 219–239. Kluwer, London (2001)
Jacinto F.M.O., Scheimberg S.: Duality for generalized equilibrium problems. Optimization 57, 795–805 (2008)
Jeyakumar V.: Asymptotic dual conditions characterizing optimality for convex programs. J. Optim. Theory Appl. 93, 153–165 (1997)
Jeyakumar, V., Dinh, N., Lee, G.M.: A new closed cone constraint qualification for convex optimization. Applied Mathematics research report AMR04/8, UNSW, Sydney, Australia (2004).
Jeyakumar V., Wu Z.Y., Lee G.M., Dinh N.: Liberating the subgradient optimality conditions from constraint qualifications. J. Glob. Optim. 34, 127–137 (2006)
Laghdir M.: Optimality conditions and Toland’s duality for a non-convex minimization problem. Matematicki Vesnik 55, 21–30 (2003)
Martinez-Legaz J.E., Sosa W.: Duality for equilibrium problems. J. Glob. Optim. 25, 311–319 (2006)
Mastroeni G.: Gap functions for equilibrium. J. Glob. Optim. 27, 411–426 (2003)
Mosco U.: Dual variational inequalities. J. Math. Anal. Appl. 40, 202–206 (1972)
Muu L.D., Nguyen V.H., Quy N.V.: Nash-Cournot oligopolistic market equilibrium models with concave cost functions. J. Glob. Optim. 41, 351–364 (2008)
Toland J.F.: Duality in non-convex optimization. J. Math. Anal. Appl. 66, 399–415 (1978)
Zalinescu C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was completed while the first author was visiting the Department of Mathematics of the University of Namur, Namur, Belgium in July-August 2007 and in August 2008.
Rights and permissions
About this article
Cite this article
Dinh, N., Strodiot, J.J. & Nguyen, V.H. Duality and optimality conditions for generalized equilibrium problems involving DC functions. J Glob Optim 48, 183–208 (2010). https://doi.org/10.1007/s10898-009-9486-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-009-9486-z