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Minimierung einer Funktion mehrerer Variablen unter Nebenbedingungen

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Optimierung

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Zusammenfassung

In diesem Kapitel wird das unrestringierte Problem aus Kap. 4,

$$\min_{{\boldsymbol{\mathrm{x}}}\in\mathbb{R}^{n}}f({\boldsymbol{\mathrm{x}}})\> ,$$
(5.1)

in zwei Schritten zur allgemeinen Problemstellung aus Kap. 2 erweitert. Erst wird dazu ein durch Gleichungsnebenbedingungen (GNB) definierter zulässiger Bereich berücksichtigt

$$\min_{{\boldsymbol{\mathrm{x}}}\in{X}}f({\boldsymbol{\mathrm{x}}}),\quad\text{wobei\quad}X=\{{\boldsymbol{\mathrm{x}}}\mid\boldsymbol{\mathrm{c}}({\boldsymbol{\mathrm{x}}})=\boldsymbol{\mathrm{0}}\}\> .$$
(5.2)

Später werden zusätzlich Ungleichungsnebenbedingungen (UNB) hinzugenommen, die den zulässigen Bereich weiter einschränken. Es gilt für den zulässigen Bereich

$$X=\{{\boldsymbol{\mathrm{x}}}\mid\boldsymbol{\mathrm{c}}({\boldsymbol{\mathrm{x}}})=\boldsymbol{\mathrm{0}},{\boldsymbol{\mathrm{h}}({\boldsymbol{\mathrm{x}}})}\leq\boldsymbol{\mathrm{0}}\}\> .$$
(5.3)

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Notes

  1. 1.

    Eigentlich bereits in der MSc-Arbeit von W. Karush [Karush(1939)] vom Jahr 1939, zusammen mit dem von H.W. Kuhn und A.W. Tucker [Kuhn and Tucker(1951)] viel später veröffentlichten Theorem, enthalten.

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Author information

Authors and Affiliations

  1. Technical University of Crete, Chania, Griechenland

    Markos Papageorgiou

  2. TU München, München, Deutschland

    Marion Leibold

  3. TU München, München, Deutschland

    Martin Buss

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  1. Markos Papageorgiou
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  2. Marion Leibold
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  3. Martin Buss
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Correspondence to Markos Papageorgiou .

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Papageorgiou, M., Leibold, M., Buss, M. (2015). Minimierung einer Funktion mehrerer Variablen unter Nebenbedingungen. In: Optimierung. Springer Vieweg, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46936-1_5

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  • DOI: https://doi.org/10.1007/978-3-662-46936-1_5

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