Abstract
It is well-known that all local minimum points of a semistrictly quasiconvex real-valued function are global minimum points. Also, any local maximum point of an explicitly quasiconvex real-valued function is a global minimum point, provided that it belongs to the intrinsic core of the function’s domain. The aim of this paper is to show that these “local min–global min” and “local max–global min” type properties can be extended and unified by a single general local–global extremality principle for certain generalized convex vector-valued functions with respect to two proper subsets of the outcome space. For particular choices of these two sets, we recover and refine several local–global properties known in the literature, concerning unified vector optimization (where optimality is defined with respect to an arbitrary set, not necessarily a convex cone) and, in particular, classical vector/multicriteria optimization.
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Acknowledgements
Nicolae Popovici’s research was supported by a grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, Project Number PN-III-P4-ID-PCE-2016-0190, within PNCDI III. The authors are grateful to the anonymous referees for the careful reading of the manuscript and especially for the constructive comments and suggestions.
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Bagdasar, O., Popovici, N. Unifying local–global type properties in vector optimization. J Glob Optim 72, 155–179 (2018). https://doi.org/10.1007/s10898-018-0656-8
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DOI: https://doi.org/10.1007/s10898-018-0656-8
Keywords
- Unified vector optimization
- Algebraic local extremal point
- Topological local extremal point
- Generalized convexity