Aus Operations-Research-Wiki
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| − | A ''quadratic programming problem'' is a programming problem with ''quadratic objective function''. Therefore, the objection function includes besides the linear term <math> c_{j}x_{j}</math> the quadratic AUSDRUCK <math>c_{jj}x_{j}^{2}</math> and/or <math>c_{ij}x_{i}x_{j}</math> for some or all <math> i\neq j </math>.
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| − | First at all we define the quadratic form, which can be expressed by the quadratic parts of a objective function <math> F\left(x\right ) </math>.
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| − | We assume that a quadratic form of a function is
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| − | <math>Q\left(x\right)= x^{t}*C*x </math>.
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| − | with: x column vektor (variable vektor) of the dimension n and C symmetric (n x n) matrix with real coefficients <math> C_{ij}</math>.
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| − | You can easily transform each quadratic part of every objective function F(x) on an nonlinear optimization problem into the form provided above.First of all all quadratic parts can be written down with real coefficients \tilde{c}_{ij}:
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| − | <math>
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| − | Q\left(x_{1},...x_{n}\right)=\tilde{c}_{11}x_{1}^{2}+2\tilde{c}_{12}x_{1}x_{2}+2\tilde{c}_{13}x_{1}x_{3}+...+2\tilde{c}_{1n}x_{1}x_{n}+
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| − | \tilde{c}_{22}x_{2}^{2}+2\tilde{c}_{23}x_{2}x_{3}+...+2\tilde{c}_{2n}x_{2}x_{n}+
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| − | \tilde{c}_{33}x_{3}^{2}+...+2\tilde{c}_{3n}x_{3}x_{n}+...
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| − | .....+\tilde{c}_{nn}x_{n}^{2}
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| − | </math>
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Version vom 25. Juni 2013, 14:14 Uhr
