Nonlinear Opt.: Basic concepts 3: Unterschied zwischen den Versionen
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Theory | Theory | ||
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Instead of the Simplex algorithm there are different solution methods which are specific for a given problem. But there is no gurantee for an optimal solution. | Instead of the Simplex algorithm there are different solution methods which are specific for a given problem. But there is no gurantee for an optimal solution. | ||
| − | + | The gereral form of a non linear optimization can be stated as: | |
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<math>max~z= f(x_{1},...,x_{n})</math> | <math>max~z= f(x_{1},...,x_{n})</math> | ||
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under the constraints that: | under the constraints that: | ||
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<math>h_{i}(x_{1},...,x_{n})\leq b_{i}\qquad i=1,...,m</math> | <math>h_{i}(x_{1},...,x_{n})\leq b_{i}\qquad i=1,...,m</math> | ||
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and: | and: | ||
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<math>x_{j}\geq 0\qquad j=1,...,n</math> | <math>x_{j}\geq 0\qquad j=1,...,n</math> | ||
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| + | By transforming the restriction | ||
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<math>h_{i}(x_{1},...,x_{n})\leq b_{i}</math> | <math>h_{i}(x_{1},...,x_{n})\leq b_{i}</math> | ||
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in | in | ||
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<math>h_{i}(x_{1},...,x_{n})-b_{i}\leq 0</math> | <math>h_{i}(x_{1},...,x_{n})-b_{i}\leq 0</math> | ||
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| + | and nominating this inequality with <math>g_{i}</math> | ||
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<math>max~z=f(x_{1},...,x_{n})</math> | <math>max~z=f(x_{1},...,x_{n})</math> | ||
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| + | in consideration of the restrictions: | ||
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<math>g_{i}(x_{1},...,x_{n})\leq 0\qquad i=1,...,m</math> | <math>g_{i}(x_{1},...,x_{n})\leq 0\qquad i=1,...,m</math> | ||
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| − | <math>\frac{\partial^2 }{\partial x^2}f(x)=4>0 \rightarrow max | + | <math>\frac{\partial^2 }{\partial x^2}f(x)=4>0 \rightarrow max</math> |
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Version vom 30. Juni 2013, 16:36 Uhr
Theory
In contrast to the Linear Optimization, which solution method is the simplex algorithm, there is no comparable solution method für Nonlinear Optimization problems. Instead of the Simplex algorithm there are different solution methods which are specific for a given problem. But there is no gurantee for an optimal solution.
The gereral form of a non linear optimization can be stated as:
under the constraints that:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): h_{i}(x_{1},...,x_{n})\leq b_{i}\qquad i=1,...,m
and:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_{j}\geq 0\qquad j=1,...,n
By transforming the restriction
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): h_{i}(x_{1},...,x_{n})\leq b_{i}
in
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): h_{i}(x_{1},...,x_{n})-b_{i}\leq 0
and nominating this inequality with
in consideration of the restrictions:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): g_{i}(x_{1},...,x_{n})\leq 0\qquad i=1,...,m
Example
Example 1 (Maximization)
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): f(x)=5x-2x^2
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial }{\partial x} f(x)=5-4x
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial^2 }{\partial x^2}f(x)=4>0 \rightarrow min
Example 1 (Minimization)
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial }{\partial x}f(x)=5+4x
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial^2 }{\partial x^2}f(x)=4>0 \rightarrow max
