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Connectedness of the Efficient Set for Strictly Quasiconcave Sets

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Abstract

Given a closed subset X in EquationSource % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaW% baaSqabeaacaWGUbaaaaaa!387D! , we show the connectedness of its efficient points or nondominated points when X is sequentially strictly quasiconcave. In the particular case of a maximization problem with n continuous and strictly quasiconcave objective functions on a compact convex feasible region of EquationSource % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaW% baaSqabeaacaWGWbaaaaaa!387F! , we deduce the connectedness of the efficient frontier of the problem. This work solves the open problem of the efficient frontier for strictly quasiconcave vector maximization problems.

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  1. Maitre de Conférences, Department of Mathematics, Limoges University, Limoges, France

    J. Benoist

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Benoist, J. Connectedness of the Efficient Set for Strictly Quasiconcave Sets. Journal of Optimization Theory and Applications 96, 627–654 (1998). https://doi.org/10.1023/A:1022616612527

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