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Optimality conditions for scalar and vector optimization problems with quasiconvex inequality constraints

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  • Ginchev Ivan

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    (Department of Economics, University of Insubria, Italy)

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    Abstract

    Let X be a real linear space, X0  X a convex set, Y and Z topological real linear spaces. The constrained optimization problem minCf(x), g(x) 2 -K is considered, where f : X0 ! Y and g : X0 ! Z are given (nonsmooth) functions, and C  Y and K  Z are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones. Key words: Vector optimization, nonsmooth optimization, quasiconvex vector functions, pseudoconvex vector functions, Dini derivatives, quasiconvex programming, Kuhn-Tucker conditions..

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    File URL: http://eco.uninsubria.it/dipeco/Quaderni/files/QF2008_5.pdf
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    Bibliographic Info

    Paper provided by Department of Economics, University of Insubria in its series Economics and Quantitative Methods with number qf0805.

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    Length: 24 pages
    Date of creation: Jun 2008
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    Handle: RePEc:ins:quaeco:qf0805

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    1. Josh Lerner, 2005. "The Scope of Open Source Licensing," Journal of Law, Economics and Organization, Oxford University Press, vol. 21(1), pages 20-56, April.
    2. West, Joel, 2003. "How open is open enough?: Melding proprietary and open source platform strategies," Research Policy, Elsevier, vol. 32(7), pages 1259-1285, July.
    3. Fabio Maria Manenti & Stefano Comino, 2010. "Dual Licensing in Open Source Software Markets," "Marco Fanno" Working Papers 0112, Dipartimento di Scienze Economiche "Marco Fanno".
    4. Lerner, Josh & Tirole, Jean, 2002. "Some Simple Economics of Open," Journal of Industrial Economics, Wiley Blackwell, vol. 50(2), pages 197-234, June.
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