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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.

##### MSC:
 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
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##### References:
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