Duality and optimality conditions for generalized equilibrium problems involving DC functions. (English) Zbl 1228.90078

Summary: 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 J. E. Martínez-Legaz and W. Sosa [J. Glob. Optim. 35, No. 2, 311–319 (2006; Zbl 1106.90074)] for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco’s dual problem [U. Mosco, J. Math. Anal. Appl. 40, 202–206 (1972; Zbl 0262.49003)] when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by F. M. O. Jacinto and S. Scheimberg [Optimization 57, No. 6, 795–805 (2008; Zbl 1152.90648)] 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 [L. Altangerel, R. I. Bot and G. Wanka, Asia-Pac. J. Oper. Res. 24, No. 3, 353–371 (2007; Zbl 1141.49303); Pac. J. Optim. 2, No. 3, 667–678 (2006; Zbl 1103.49016)] 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 [N. Dinh, T. T. A. Nghia and G. Vallet, Optimization 59, No. 3–4, 541–560 (2010; Zbl 1218.90155); J. Convex Anal. 15, No. 2, 235–262 (2008; Zbl 1145.49016)], Fenchel-Lagrange and Lagrange dualities for convex problems as in [Antangerel et al., loc. cit.; R. I. Bot and G. Wanka, Nonlinear Anal., Theory Methods Appl. 64, No. 12, A, 2787–2804 (2006; Zbl 1087.49026); V. Jeyakumar, N. Dinh and G. M. Lee, “A new closed cone constraint qualification for convex optimization”, Applied mathematics research report AMR04/8, University of New South Wales, Sidney, Australia (2004)]. Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems.


90C26 Nonconvex programming, global optimization
90C46 Optimality conditions and duality in mathematical programming
49N15 Duality theory (optimization)
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
Full Text: DOI


[1] 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)
[2] Antangerel L., Bot R.I., Wanka G.: On gap functions for equilibrium problems via Fenchel duality. Pac. J. Optim. 2, 667–678 (2006) · Zbl 1103.49016
[3] 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)
[4] Auslender A.: Optimisation. Méthodes Numériques. Masson, Paris (1976)
[5] Bigi G., Castellani M., Kassay G.: A dual view of equilibrium problems. J. Math. Anal. Appl. 342, 17–26 (2008) · Zbl 1155.90021
[6] Blum E., Oettli W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 127–149 (1994) · Zbl 0888.49007
[7] Bot, R.I., Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Anal, to appear.
[8] Burachik R.S., Jeyakumar V.: A new geometric condition for Fenchel’s duality in infinite dimensional spaces. Math. Program. 104(B), 229–233 (2005) · Zbl 1093.90077 · doi:10.1007/s10107-005-0614-3
[9] Burachik R.S., Jeyakumar V.: A dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12, 279–290 (2005) · Zbl 1098.49017
[10] 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) · Zbl 1126.90059 · doi:10.1051/cocv:2007027
[11] 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) · Zbl 1114.90083 · doi:10.1007/s10957-004-1712-8
[12] 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.) · Zbl 1226.90102
[13] 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\(\sim\)content=g770174694\(\sim\)db=all?stem=3#messages · Zbl 1218.90155
[14] 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) · Zbl 1145.49016
[15] Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for extended Farkas’ lemmas and Lagrangian dualities in convex infinite programming (Submitted). · Zbl 1206.90198
[16] Fukushima, M.: A class of gap functions for quasi-variational inequality problems (Preprint) · Zbl 1170.90487
[17] 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) · Zbl 0735.90056
[18] Jacinto F.M.O., Scheimberg S.: Duality for generalized equilibrium problems. Optimization 57, 795–805 (2008) · Zbl 1152.90648 · doi:10.1080/02331930701761458
[19] Jeyakumar V.: Asymptotic dual conditions characterizing optimality for convex programs. J. Optim. Theory Appl. 93, 153–165 (1997) · Zbl 0901.90158 · doi:10.1023/A:1022606002804
[20] 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).
[21] 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) · Zbl 1131.90069 · doi:10.1007/s10898-006-9003-6
[22] Laghdir M.: Optimality conditions and Toland’s duality for a non-convex minimization problem. Matematicki Vesnik 55, 21–30 (2003) · Zbl 1051.49020
[23] Martinez-Legaz J.E., Sosa W.: Duality for equilibrium problems. J. Glob. Optim. 25, 311–319 (2006) · Zbl 1106.90074 · doi:10.1007/s10898-005-3840-6
[24] Mastroeni G.: Gap functions for equilibrium. J. Glob. Optim. 27, 411–426 (2003) · Zbl 1061.90112 · doi:10.1023/A:1026050425030
[25] Mosco U.: Dual variational inequalities. J. Math. Anal. Appl. 40, 202–206 (1972) · Zbl 0262.49003 · doi:10.1016/0022-247X(72)90043-1
[26] 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) · Zbl 1146.91029 · doi:10.1007/s10898-007-9243-0
[27] Toland J.F.: Duality in non-convex optimization. J. Math. Anal. Appl. 66, 399–415 (1978) · Zbl 0403.90066 · doi:10.1016/0022-247X(78)90243-3
[28] Zalinescu C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)
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