## On the Number of Criteria Needed to Decide Pareto Optimality

• In this paper we address the question of how many objective functions are needed to decide whether a given point is a Pareto optimal solution for a multicriteria optimization problem. We extend earlier results showing that the set of weakly Pareto optimal points is the union of Pareto optimal sets of subproblems and show their limitations. We prove that for strictly quasi-convex problems in two variables Pareto optimality can be decided by consideration of at most three objectives at a time. Our results are based on a geometric characterization of Pareto, strict Pareto and weak Pareto solutions and Helly's Theorem. We also show that a generalization to quasi-convex objectives is not possible, and state a weaker result for this case. Furthermore, we show that a generalization to strictly Pareto optimal solutions is impossible, even in the convex case.

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Verfasserangaben: Matthias Ehrgott, Stefan Nickel urn:nbn:de:hbz:386-kluedo-10682 Report in Wirtschaftsmathematik (WIMA Report) (61) Preprint Englisch 2000 2000 Technische Universität Kaiserslautern 29.08.2000 Multicriteria optimization ; Pareto optimality ; number of objectives; strictly quasi-convex functions Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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