Abstract
LetF be a family of real-valued maps onR n, and letY be a subset ofR n. Denote byS(Y|F) the set of ally* ∈Y such that, for somef ∈F,f(y)≥f(y*) for ally inY. Let us say thatF is a scalarization family if, for any subsetY,S(Y|F) is equal to the set of properly efficient points inY. General conditions forF to be a scalarization family were given in Ref. 1. However, scalarization families must contain nondifferentiable functions. In this note, it is shown that, if the condition of Ref. 1 which forces nondifferentiability is dropped, thenS(Y|F) is dense in the set of properly efficient points.
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Communicated by P. L. Yu
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Gearhart, W.B. Families of differentiable scalarization functions. J Optim Theory Appl 62, 321–332 (1989). https://doi.org/10.1007/BF00941061
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DOI: https://doi.org/10.1007/BF00941061