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Conditions for ε-Pareto Solutions in Multiobjective Optimization 123We denote by AE(f, S, ‖ ‖, ε) and WAE(f, S, ‖ ‖, ε) the sets of ε-Pareto solutions and weakε-Pareto solutions of (1), respectively. Let us observe that these sets depend on the norm ‖ ‖.Moreover, it is clear that AE(f, S, ‖ ‖, ε) ⊂ WAE(f, S, ‖ ‖, ε) for each ε ≥ 0 and AE(f, S, ‖ ‖, 0) =E(f, S), WAE(f, S, ‖ ‖, 0) = WE(f, S).Next, we show under different hypotheses that Tanaka’s concept is a metrically consistentε-efficiency notion.Theorem 4. Let f : X → R p be a continuous map at x 0 ∈ S and let (ε n ) ⊂ R + , (x n ) ⊂ S be suchthat ε n ↓ 0 and x n → x 0 .1. If x n ∈ AE(f, S, ‖ ‖, ε n ) for each n, then x 0 ∈ WE(f, S).2. If (f(x n )) is a nonincreasing sequence, i.e.,f(x m ) ∈ f(x n ) − R p + ,and x n ∈ AE(f, S, ‖ ‖, ε n ) for each n, then x 0 ∈ E(f, S).∀ m > n3. Suppose that f(S) is externally stable with respect to the efficient set:f(S) ⊂ f(E(f, S)) + R p + .If x n ∈ AE(f, S, ‖ ‖, ε n ) for each n, then f(x 0 ) ∈ cl(f(E(f, S))).3. Multiplier rules for ε-Pareto solutionsIn this section, multiplier rules for ε-Pareto solutions of (1) are proved under convexity hypotheses.So, let us suppose that f i and g j are p + m continuous convex functionals and Gis a convex set. We use Ker(A) and Ker(A) ⊥ to denote the kernel of A and the orthogonalcomplement of Ker(A). The topological dual space of X is denoted by X ∗ . We write I G for theindicator functional of the set G and ‖ ‖ 1 for the l 1 norm in R p .To attain our objective, we use the ε-subdifferential of a proper convex functional and apenalized scalarization process.Definition 5. Let h : X → R ∪ {∞} be a convex proper functional, x 0 ∈ dom(h) and ε ≥ 0. Theε-subdifferential of h at x 0 is the set ∂h ε (x 0 ) defined by∂ ε h(x 0 ) = {x ∗ ∈ X ∗ : h(x) ≥ h(x 0 ) − ε + 〈x ∗ , x − x 0 〉, ∀ x ∈ X}.Let us recall that the subdifferential ∂h(x 0 ) of h at x 0 in the sense of Convex Analysis isobtained taking ε = 0 in Definition 5.Theorem 6. If Q∩int(G) ≠ ∅ and x 0 ∈ WAE(f, S, ‖ ‖ 1 , ε) then there exists (τ, ν, α) ∈ R p ×R m ×Rand multipliers (λ, γ, µ) ∈ R p × R × R m such that0 ∈p∑∂ τi ((λ i + γ)f i )(x 0 ) +i=1(τ, ν, α, λ, γ, µ) ≥ 0, (3)p∑m∑λ i + γ + µ j = 1, (4)i=1j=1m∑∂ νj (µ j g j )(x 0 ) + Ker(A) ⊥ + ∂ α I G (x 0 ), (5)j=1p∑τ i +i=1m∑ν j − γε + α ≤j=1m∑µ j g j (x 0 ). (6)Theorem 7. Consider x 0 ∈ S. If there exists (τ, ν, α, λ, γ, µ) ∈ R p × R m × R × R p × R × R mverifying conditions (3)-(6) with strict inequality in (6), then x 0 ∈ AE(f, S, ‖ ‖ 1 , ε).j=1