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122 C. Gutiérrez, B. Jiménez, and V. Novosome properties are established. Then, we describe a general method to convert a constrainedmultiobjective optimization problem into a scalar optimization problem without inequalityconstraints in such a way that ε-Pareto solutions for the first problem are approximate solutionsfor the second problem. In Section 3, Fritz John and Kuhn-Tucker necessary and sufficientconditions for ε-Pareto solutions are proved via the ε-subdifferential and the scalarizationmethod developed in Section 2. In obtaining these conditions, convexity hypothesesare assumed. In Section 4, a sufficient condition for ε-Pareto solutions is proved through approximatesolutions of a scalar Lagrangian functional. From this result, it is possible to obtainapproximate metrically consistent solutions of constrained multiobjective optimization problemsvia suboptimal solutions of unconstrained scalar optimization problems.2. Metrically consistent ε-Pareto solutionsLet X be a normed space and let us fix p + m functionals f i , g j : X → R, i = 1, 2, . . . , p,j = 1, 2, . . . , m, a continuous linear map A : X → R k and a nonempty set G ⊂ X. In thesequel, the following constrained multiobjective optimization problem is consideredwhere f = (f 1 , f 2 , . . . , f p ) : X → R p , S = K ∩ Q ∩ G,Min{f(x) : x ∈ S}, (1)K = {x ∈ X : g j (x) ≤ 0, j = 1, 2, . . . , m},Q = {x ∈ X : Ax = b},and b ∈ R k .In solving (1), the componentwise partial order in the final space is assumed. Let us denoteby cl(M) and int(M) the closure and interior of a set M, respectively. The nonnegative orthantin R p is denoted by R p + and we write the set int( R p )+ as RpDefinition 1. A point x 0 ∈ S is a Pareto (resp. weak Pareto) solution of (1) if(resp. (f(x 0 ) − R p ++ ) ∩ f(S) = ∅).++ .(f(x 0 ) − R p + \{0}) ∩ f(S) = ∅The set of Pareto solutions and weak Pareto solutions of (1) will be denoted by E(f, S) andWE(f, S), respectively. It is clear that E(f, S) ⊂ WE(f, S).Consider ε ≥ 0 and the scalar optimization problemMin{h(x) : x ∈ M}, (2)where h : X → R and M ⊂ X, M ≠ ∅. In the following definition, the well-known notion ofapproximate (suboptimal) solution of (2) is recalled.Definition 2. A point x 0 ∈ M is an ε-solution of (2) ifh(x 0 ) − ε ≤ h(x), ∀ x ∈ M.The set of ε-solutions of (2) is denoted by AMin(h, M, ε).Let B ⊂ R p be the unit closed ball of R p defined by a norm ‖ ‖. Next, we recall a concept introducedby Tanaka [14], which extends Definition 2 to multiobjective optimization problems.Definition 3. A point x 0 ∈ S is an ε-Pareto (resp. weak ε-Pareto) solution of (1) if(resp. (f(x 0 ) − ((εB) c ∩ R p ++ )) ∩ f(S) = ∅).(f(x 0 ) − ((εB) c ∩ R p + )) ∩ f(S) = ∅