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On global minima of semistrictly quasiconcave functions

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Abstract

This paper studies the global behaviour of semistrictly quasiconcave functions with (possibly) nonconvex domain in the presence of global minima. We mainly present necessary conditions for the existence of global minima of a semistrictly quasiconcave real-valued function f with domain \({K\subset {\mathbb{R}}^{n}}\) , and we show how the geometric structure of its graph and the cardinality of its range depend on the location of global minimum points. Our main result states that if a global minimum of f is achieved in the algebraic interior of K, then f can attain at the most n + 1 distinct function values, and the graph of f has a simple structure determined by a sequence of nested affine subspaces such that, essentially, f is constant on the set difference of each pair of successive affine subspaces.

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References

  1. Avriel M., Diewert W.E., Schaible S., Zang I.: Generalized Concavity. Plenum Press, New York (1988)

    MATH  Google Scholar 

  2. Bereanu B.: On the global minimum of a quasi-concave functional. Arch. Math. XXV, 391–393 (1974)

    Article  MathSciNet  Google Scholar 

  3. Cambini A., Martein L.: Generalized convexity and optimality conditions in scalar and vector optimization. In: Hadjisavvas, N., Komlósi, S., Schaible, S.(eds) Handbook of Generalized Convexity and Generalized Monotonicity., pp. 151–193. Springer Science + Business Media, New York (2005)

    Google Scholar 

  4. Daniilidis A., Hadjisavvas N.: Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions. J. Optim. Theory Appl. 102, 525–536 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Haberl J.: Maximization of generalized convex functionals in locally convex spaces. J. Optim. Theory Appl. 121, 327–359 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Holmes R.B.: Geometric Functional Analysis and its Applications. Springer, New York (1975)

    MATH  Google Scholar 

  7. Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction toGlobalOptimization. Kluwer,Dordrecht (2000)

    MATH  Google Scholar 

  8. Horst, R., Pardalos, P.M. (eds.): Handbook of Global Optimization. Kluwer, Dordrecht (1995)

    MATH  Google Scholar 

  9. Karamardian S.: Strictly quasi-convex (concave) functions and duality in mathematical programming. J. Math. Anal. Appl. 20, 344–358 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mangasarian, O.L.: Nonlinear Programming. SIAM, Philadelphia (1994) (original work published 1969)

  11. Thompson W.A., Parke D.W.: Some properties of generalized concave functions. Oper. Res. 21, 305–313 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  12. van Tiel J.: Convex Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

Download references

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  1. School of Medical Information Technology, Carinthia University of Applied Sciences, Primoschgasse 10, 9020, Klagenfurt, Austria

    Josef Haberl

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  1. Josef Haberl
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Correspondence to Josef Haberl.

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Haberl, J. On global minima of semistrictly quasiconcave functions. Optim Lett 3, 387–396 (2009). https://doi.org/10.1007/s11590-009-0118-9

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  • DOI: https://doi.org/10.1007/s11590-009-0118-9

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