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Optimization on directionally convex sets


  • Vladimir Naidenko



Directional convexity generalizes the concept of classical convexity. We investigate OC-convexity generated by the intersections of C-semispaces that efficiently approximates directional convexity. We consider the following optimization problem in case of the direction set of OC-convexity being infinite. Given a compact OC-convex set A, maximize a linear form L subject to A. We prove that there exists an OC-extreme solution of the problem. We introduce the notion of OC-quasiconvex function. Ii is shown that if O is finite then the constrained maximum of an OC-quasiconvex function on the set A is attained at an OC-extreme point of A. We show that the OC-convex hull of a finite point set represents the union of a finite set of polytopes in case of the direction set being finite. Copyright Springer-Verlag 2009

Suggested Citation

  • Vladimir Naidenko, 2009. "Optimization on directionally convex sets," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 17(1), pages 55-63, March.
  • Handle: RePEc:spr:cejnor:v:17:y:2009:i:1:p:55-63 DOI: 10.1007/s10100-008-0074-y

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    References listed on IDEAS

    1. Solymosi, T. & Raghavan, T.E.S. & Tijs, S.H., 2003. "Bargaining sets and the core in permutation games," Other publications TiSEM a14f6955-62c4-4bf0-8a54-f, Tilburg University, School of Economics and Management.
    2. Curiel, I. & Tijs, S.H., 1986. "Assignment games and permutation games," Other publications TiSEM c9a47c3b-28d3-4874-b0a2-f, Tilburg University, School of Economics and Management.
    3. Daniel Granot, 2010. "The reactive bargaining set for cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 163-170, March.
    4. Tijs, S.H. & Parthasarathy, T. & Potters, J.A.M. & Rajendra Prasad, V., 1984. "Permutation games : Another class of totally balanced games," Other publications TiSEM a7edfa18-6224-4be3-b677-5, Tilburg University, School of Economics and Management.
    5. Tamás Solymosi, 2002. "The bargaining set of four-person balanced games," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 1-11.
    6. Solymosi, Tamas & Raghavan, T. E. S. & Tijs, Stef, 2005. "Computing the nucleolus of cyclic permutation games," European Journal of Operational Research, Elsevier, vol. 162(1), pages 270-280, April.
    7. Quint, Thomas, 1996. "On One-Sided versus Two-Sided Matching Games," Games and Economic Behavior, Elsevier, vol. 16(1), pages 124-134, September.
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    Directional convexity; Optimization;


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