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Proceedings of GO 2005, pp. 121 – 125.Conditions for ε-Pareto Solutionsin Multiobjective OptimizationC. Gutiérrez, 1 B. Jiménez, 2 and V. Novo 31 Universidad de Valladolid, Valladolid, Spain, cesargv@mat.uva.es2 Universidad de Salamanca, Salamanca, Spain, bjimen1@encina.pntic.mec.es3 Universidad Nacional de Educación a Distancia, Madrid, Spain, vnovo@ind.uned.esAbstractKeywords:In this paper, approximate Pareto solutions of nondifferentiable constrained multiobjective optimizationproblems are studied via a metrically consistent ε-efficient concept introduced by Tanaka[14]. Necessary and sufficient conditions for these solutions are obtained from a nonconvex penalizedscalarization process. Necessary conditions are provided in convex multiobjective optimizationproblems through Kuhn-Tucker multiplier rules. Sufficient conditions are obtained via Kuhn-Tuckermultiplier rules under convexity hypotheses and via approximate solutions of a scalar Lagrangianproblem for nonconvex multiobjective optimization problems.Multiobjective optimization, ε-Pareto solution, scalarization, ε-subdifferential.1. IntroductionDuring the last decades, interest in approximate solutions or ε-efficient solutions (ε-Paretosolutions in the Paretian context) of vector optimization problems is growing, since these solutionsexist under very mild hypotheses and they are obtained by a lot of usual resolutionmethods (for example, by iterative algorithms, heuristic methods, etc.).The first and most popular ε-efficient concept was introduced by Kutateladze [8] and hasbeen used to establish approximate Kuhn-Tucker type conditions and approximate dualitytheorems [2, 4, 9–13, 15]. However, Kutateladze’s ε-efficiency concept gives approximate solutionswhich are not metrically consistent, i.e., it is possible to obtain feasible points (x n ), x 0such that their objective values verify f(x n ) → f(x 0 ), x n is an ε n -efficient solution for each n,ε n → 0, and f(x 0 ) is far from the optimal value set.In [1, 4–7, 15, 18], various metrically consistent ε-efficient concepts based on a previouslyfixed scalar functional have been studied. However, there are a lot of problems for which isnot possible to choose any previous scalar functional and so several other metrically consistentnotions have been introduced without consider any additional scalar functional (see, forexample, the concepts defined by White [16] and Tanaka [14]).Classical conditions for efficient solutions via multiplier rules, Lagrangian functionals andsaddlepoint theorems must be extended to ε-Pareto solutions in order to develop new andbetter resolution methods. In [2, 3, 9–12, 17] some results have been obtained following thisline, but using not metrically consistent ε-Pareto concepts. In this work, Tanaka’s ε-Paretonotion is analyzed from these points of view in order to extend those classical conditions to ametrically consistent ε-Pareto concept without consider any additional scalar functional.In Section 2, the nondifferentiable constrained multiobjective optimization problem is presentedand some notations are fixed. Moreover, the Tanaka’s ε-Pareto concept is recalled and