Abstract
In a recent paper by Li (Ref. 1), a scheme was proposed to convexify an efficient frontier for a vector optimization problem by rescaling each component of the vector objective functions by its p-power. For sufficiently large p, it was shown that the transformed efficient frontier is cone-convex; hence, the usual linear scalarization (or supporting hyperplane) method can be used to find the efficient solutions. An outstanding question remains: What is the minimum value of p such that the efficient frontier can be convexified? In this note, we answer the above question by deriving some theoretical lower bounds for p.
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References
Li, D., Convexification of a Noninferior Frontier, Journal of Optimization Theory and Applications, Vol. 88, pp. 177–196, 1996.
Goh, C. J., and Caughey, R. K., On Stability Problems Caused by Finite Actuator Dynamics in the Collocated Control of Large Space Structures, International Journal of Control, Vol. 41, pp. 787–802, 1985.
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Goh, C.J., Yang, X.Q. Convexification of a Noninferior Frontier. Journal of Optimization Theory and Applications 97, 759–768 (1998). https://doi.org/10.1023/A:1022654528902
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DOI: https://doi.org/10.1023/A:1022654528902