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γ-Subdifferential and γ-convexity of functions on the real line


In this paper the γ-subdifferential and γ-convexity of real-valued functions on the real line are introduced. By means of the γ-subdifferential, a new necessary condition for global minima (or maxima) is formulated which many local minima (or maxima) cannot satisfy. The γ-convexity is used to state sufficient conditions for global minima. The class of γ-convex functions is relatively large. For example, there are γ-convex functions which are not continuous anywhere. Nevertheless, a γ-local minimum of a γ-convex function is always a global minimum. Furthermore, if a γ-convex function attains its global minimum, then it does so near the boundary of its domain.

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Communicated by J. Stoer

This research was supported by the Alexander von Humboldt Foundation.

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Phú, H.X. γ-Subdifferential and γ-convexity of functions on the real line. Appl Math Optim 27, 145–160 (1993).

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Key words

  • Subdifferential
  • Convex
  • Quasi-convex
  • Optimization
  • Minimum
  • Maximum
  • Necessary condition
  • Sufficient condition

AMS classification

  • 46G05
  • 49B27
  • 52A01
  • 65K05
  • 90C25
  • 90C48