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Generalized semi-infinite optimization: A first order optimality condition and examples

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Abstract

We consider a generalized semi-infinite optimization problem (GSIP) of the form (GSIP) min{f(x)‖xεM}, where M={x∈ℝn|hi(x)=0i=l,...m, G(x,y)⩾0, y∈Y(x)} and all appearing functions are continuously differentiable. Furthermore, we assume that the setY(x) is compact for allx under consideration and the set-valued mappingY(.) is upper semi-continuous. The difference with a standard semi-infinite problem lies in thex-dependence of the index setY. We prove a first order necessary optimality condition of Fritz John type without assuming a constraint qualification or any kind of reduction approach. Moreover, we discuss some geometrical properties of the feasible setM.

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Authors and Affiliations

  1. Department of Mathematics (C), RWTH Aachen, 52056, Aachen, Germany

    H. Th. Jongen & O. Stein

  2. Institute of Applied Mathematics II, University of Erlangen-Nürnberg, 91058, Erlangen, Germany

    J. -J. Rückmann

Authors
  1. H. Th. Jongen
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  2. J. -J. Rückmann
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  3. O. Stein
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Corresponding author

Correspondence to O. Stein.

Additional information

This work was partially supported by the “Deutsche Forschungsgemeinschaft” through the Graduiertenkolleg “Mathematische Optimierung” at the University of Trier.

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Jongen, H.T., Rückmann, J.J. & Stein, O. Generalized semi-infinite optimization: A first order optimality condition and examples. Mathematical Programming 83, 145–158 (1998). https://doi.org/10.1007/BF02680555

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  • DOI: https://doi.org/10.1007/BF02680555

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