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Supremizers of inner γ-convex functions


A real-valued function f defined on a convex subset D of some normed linear space is said to be inner γ-convex w.r.t. some fixed roughness degree γ  >  0 if there is a \(\nu \in]0, 1]\) such that \({\rm sup}_{\lambda\in[2,1+1/\nu]} \left(f((1 - \lambda)x_0 + \lambda x_1) - (1 - \lambda) f (x_0)-\right. \left.\lambda f(x_1)\right) \geq 0\) holds for all \(x_0, x_1 \in D\) satisfying ||x 0 − x 1||  =  νγ and \(-(1/\nu)x_0+(1+1/\nu)x_1\in D\) . This kind of roughly generalized convex functions is introduced in order to get some properties similar to those of convex functions relative to their supremum. In this paper, numerous properties of their supremizers are given, i.e., of such \(x^* \in D\) satisfying lim \({\rm sup}_{x \to x^*}f(x) = {\rm sup}_{x \in D} f(x)\) . For instance, if an upper bounded and inner γ-convex function, which is defined on a convex and bounded subset D of some inner product space, has supremizers, then there exists a supremizer lying on the boundary of D relative to aff D or at a γ-extreme point of D, and if D is open relative to aff D or if dim D  ≤  2 then there is certainly a supremizer at a γ-extreme point of D. Another example is: if D is an affine set and \(f : D \to {\mathbb{R}}\) is inner γ-convex and bounded above, then \({\rm sup}_{x'\in \bar B(x,\gamma/2)\cap D}f(x')= \sup_{x'\in D}f(x')\) for all \(x \in D\) , and if 2 ≤  dim D  <  ∞ then each \(x \in D\) is a supremizer of f.

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  1. Hartwig H (1992). Local boundedness and continuity of generalized convex functions. Optimization 26: 1–13

    Article  MATH  MathSciNet  Google Scholar 

  2. Hu TC, Klee V and Larman D (1989). Optimization of globally convex functions. SIAM J Control Optim 27: 1026–1047

    Article  MATH  MathSciNet  Google Scholar 

  3. Phu HX (1993). γ-Subdifferential and γ-convexity of functions on the real line. Appl Math Optim 27: 145–160

    Article  MATH  MathSciNet  Google Scholar 

  4. Phu HX (1994). Representation of bounded convex sets by rational convex hull of its γ-extreme points. Numer Funct Anal Optim 15: 915–920

    Article  MATH  MathSciNet  Google Scholar 

  5. Phu HX (1995). γ-Subdifferential and γ-convex functions on a normed space. J Optim Theory Appl 85: 649–676

    Article  MATH  MathSciNet  Google Scholar 

  6. Phu HX (1997). Six kinds of roughly convex functions. J Optim Theory Appl 92: 357–375

    Article  MATH  MathSciNet  Google Scholar 

  7. Phu HX (2007a) Inner γ-convex functions in normed spaces, E-Preprint 2007/01/01, Hanoi Institute of Mathematics

  8. Phu HX (2007b). Outer γ-convexity and inner γ-convexity of disturbed functions. Vietnam J Math 35: 107–119

    MathSciNet  Google Scholar 

  9. Rockafellar RT (1970). Convex analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

  10. Söllner B (1991) Eigenschaften γ-grobkonvexer Mengen und Funktionen. Diplomarbeit, Universität Leipzig

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Correspondence to Hoang Xuan Phu.

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Phu, H.X. Supremizers of inner γ-convex functions. Math Meth Oper Res 67, 207–222 (2008).

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  • Generalized convexity
  • Rough convexity
  • Inner γ-convex function
  • Supremizer
  • γ-Extreme point

Mathematics Subject Classification (2000)

  • 52A01
  • 52A41
  • 90C26