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γ-Subdifferential and γ-convexity of functions on a normed space


In this paper, the γ-subdifferential ∂γ is introduced for investigating the global behavior of real-valued functions on a normed spaceX. Iff: DX → ℝ attains its global minimum onD atx *, then 0∈∂γ f(x *). This necessary condition always holds, even iff is not continuous orx * is at the boundary of its domain. Nevertheless, it is useful because, by choosing a suitable γ∈ℝ+, many local minima cannot satisfy this necessary condition. For the sufficient conditions, the so-called γ-convex functions are defined. The class of these functions is rather large. For example, every periodic function on the real line is a γ-convex function. There are γ-convex functions which are not continuous everywhere. Every function of bounded variation can be represented as the difference of two γ-convex functions. For all that, γ-convex functions still have properties similar to those of convex functions. For instance, each γ-local minimizer off is at the same time a global one. Iff attains its global minimum onD, then it does so at least at one point of its γ-boundary.

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Communicated by R. Bulirsch

This research was supported by the Alexander von Humboldt Foundation. The author thanks Professors R. Bulirsch, K. H. Hoffmann, and H. G. Bock for inviting him to Munich and Augsburg where this research was done.

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Phu, H.X. γ-Subdifferential and γ-convexity of functions on a normed space. J Optim Theory Appl 85, 649–676 (1995).

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Key Words

  • Subdifferentials
  • convex functions
  • quasiconvex functions
  • optimization
  • minima
  • maxima
  • necessary conditions
  • sufficient conditions