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124 C. Gutiérrez, B. Jiménez, and V. Novo4. Conditions for ε-Pareto solutions via a LagrangianfunctionalIn this section, a sufficient condition for ε-Pareto solutions of (1) is established through approximatesolutions of a unconstrained scalar optimization problem obtained from (1) via aLagrangian functional.In the sequel, we extend the final space to R p ∪{±∞} and we assume that the usual algebraicand ordering properties hold. Let L : X ×R p ×R m ×R k → R∪{±∞} be the Lagrangian scalarfunctional defined by⎧⎨L(x, λ, µ, ρ) =⎩∞ if x /∈ G〈λ, f(x)〉 + 〈µ, g(x)〉 + 〈ρ, A(x)〉 if x ∈ G, µ ∈ R m +−∞ if x ∈ G, µ /∈ R m +.Fixed (λ, µ, ρ) ∈ R p ×R m ×R k , let us consider the following Lagrangian scalar optimizationproblem of (1):Min{L λ,µ,ρ (x) : x ∈ X}, (7)where L λ,µ,ρ : X → R ∪ {±∞} is the functional defined for all x ∈ X by L λ,µ,ρ (x) =L(x, λ, µ, ρ). Next, we obtain a sufficient condition for ε-Pareto solutions of (1) through approximatesolutions of (7).Theorem 8. Let (λ 0 , µ 0 , ρ 0 ) ∈ R p ++ ×Rm + ×R k and x 0 ∈ S be such that x 0 ∈ AMin(L λ0 ,µ 0 ,ρ 0, X, ε).Then x 0 ∈ AE(f, S, ‖ ‖ 1 , ε 0 ), whereAcknowledgmentsε 0 = (ε − 〈µ 0 , g(x 0 )〉)/ min1≤i≤p {λ i}.This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), projectBFM2003-02194.References[1] D. Dentcheva and S. Helbig. On variational principles, level sets, well-posedness, and ε-solutions in vectoroptimization. J. Optim. Theory Appl., 89(2):325–349, 1996.[2] J. Dutta and V. Vetrivel. On approximate minima in vector optimization. Numer. Funct. Anal. Optim.,22(7&8):845–859, 2001.[3] M. G. Govil and A. Mehra. ε-optimality for multiobjective programming on a Banach space. European J.Oper. Res., 157:106–112, 2004.[4] C. Gutiérrez. Condiciones de ε-Eficiencia en Optimización Vectorial. PhD thesis, Universidad Nacional de Educacióna Distancia, Madrid, 2004.[5] C. Gutiérrez, B. Jiménez, and V. Novo. A chain rule for ε-subdifferentials with applications to approximatesolutions in convex Pareto problems. J. Math. Anal. Appl., In press, 2005.[6] C. Gutiérrez, B. Jiménez, and V. Novo. Multiplier rules and saddle-point theorems for Helbig’s approximatesolutions in convex Pareto problems. J. Global Optim., 32(3), 2005.[7] S. Helbig and D. Pateva. On several concepts for ε-efficiency. OR Spektrum, 16:179–186, 1994.[8] S. S. Kutateladze. Convex ε-programming. Soviet Math. Dokl., 20(2):391–393, 1979.[9] J. C. Liu. ε-duality theorem of nondifferentiable nonconvex multiobjective programming. J. Optim. TheoryAppl., 69(1):153–167, 1991.[10] J. C. Liu. ε-Pareto optimality for nondifferentiable multiobjective programming via penalty function. J. Math.Anal. Appl., 198:248–261, 1996.