Abstract
This paper considers six kinds of roughly convex functions, namely: δ-convex, midpoint δ-convex, ρ-convex, γ-convex, lightly γ-convex, and midpoint γ-convex functions. The relations between these concepts are presented. It is pointed out that these roughly convex functions have two optimization properties: each r-local minimizer is a global minimizer, and if they assume their maximum on a bounded convex domain D (in a Hilbert space), then they do so at least at one r-extreme point of D, where r denotes the roughness degree of these functions. Furthermore, analytical properties are investigated, such as boundedness, continuity, and conservation properties.
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Phu, H.X. Six Kinds of Roughly Convex Functions. Journal of Optimization Theory and Applications 92, 357–375 (1997). https://doi.org/10.1023/A:1022611314673
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DOI: https://doi.org/10.1023/A:1022611314673