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On the Structure of the Weakly Efficient Set for Quasiconvex Vector Minimization

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Abstract

We investigate conditions under which the weakly efficient set for minimization of m objective functions on a closed and convex \(X\subset \mathbb R^d\) (\(m>d\)) is fully determined by the weakly efficient sets for all n-objective subsets for some \(n<m\). For quasiconvex functions, it is their union with \(n=d+1\). For lower semi-continuous explicitly quasiconvex functions, the weakly efficient set equals the linear enclosure of their union with \(n=d\), as soon as it is bounded. Sufficient conditions for the weakly efficient set to be bounded or unbounded are also investigated.

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Notes

  1. It should be noted that several recent authors also call ‘strictly quasiconvex’ what is called explicitly quasiconvex here, e.g. [6,7,8]. The two notions are quite different, since exqc allows for level sets \(L_=(f,r)\) containing a nontrivial line segment, which is excluded for sqc. Our naming follows rather the proposal of [8] that combines those in [9] for ssqc and in [10] for exqc (but without requiring continuity). See [9] for more on these (and other) naming conventions and confusions.

  2. With an unfortunate erroneous substitution of \(\cap \) instead of \(\cup \) in the definition of B(x) on p217

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  1. Prof. Em. BUTO, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium

    Frank Plastria

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  1. Frank Plastria
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Plastria, F. On the Structure of the Weakly Efficient Set for Quasiconvex Vector Minimization. J Optim Theory Appl 184, 547–564 (2020). https://doi.org/10.1007/s10957-019-01608-6

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