Nonlinear Opt.: Basic concepts 2: Unterschied zwischen den Versionen
| [unmarkierte Version] | [unmarkierte Version] |
Cabel (Diskussion | Beiträge) |
|||
| Zeile 2: | Zeile 2: | ||
== Theory == | == Theory == | ||
| − | In contrast to '''linear programming''' | + | In contrast to the '''linear programming''' where we have the [http://en.wikipedia.org/wiki/Simplex_algorithm simplex algorithm] to solve this there is not an universal algorithm for solving a '''non-linear problem'''. |
| Zeile 17: | Zeile 17: | ||
<math>f(x) = x^2 - 8x </math> | <math>f(x) = x^2 - 8x </math> | ||
| − | '''Step 1:''' | + | '''Step 1:''' The necessary condition for solving this problem is to derive the function and set this derivative equal to zero. |
| − | <math>f'(x) = 2x - 8 ~ | + | <math>f'(x) = 2x - 8 ~~=> ~2x - 8 = 0 ~<=>~ x = 4</math> |
| + | '''Step 2:''' Now you must check the sufficient condition; if you have a minimum, the second derivative of the function has to be greater than zero, for a maximum less than zero. | ||
| + | |||
| + | <math>f''(x) = 2 ~> 0 ~~=> minimum </math> | ||
Version vom 22. Juni 2013, 16:19 Uhr
In a non-linear problem there is either a non-linear objective function and no restrictions or a non-linear objective function and linear/non-linear restrictions or a linear objective function and non-linear restrictions.
Inhaltsverzeichnis
Theory
In contrast to the linear programming where we have the simplex algorithm to solve this there is not an universal algorithm for solving a non-linear problem.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): restrictions(non-linear): g_i(x) \begin{Bmatrix} >= \\ = \\ <= \end{Bmatrix}~0 ~~~~~~~~~g_i(x) ~:= ~f_i(x) ~- ~b ~<= ~0 ~~~~~~(for~ i ~= ~1,...,n)
Examples
Example 1 (non-linear objective function, no restrictions)
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): f(x) = x^2 - 8x
Step 1: The necessary condition for solving this problem is to derive the function and set this derivative equal to zero.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): f'(x) = 2x - 8 ~~=> ~2x - 8 = 0 ~<=>~ x = 4
Step 2: Now you must check the sufficient condition; if you have a minimum, the second derivative of the function has to be greater than zero, for a maximum less than zero.
Example 2 (non-linear objective function, linear restrictions)
Example 3 (non-linear objective function, non-linear restrictions)
Example 4 (linear objective function, non-linear restrictions)
Sources
Internet sources
Literature
- Prof. Dr. Oliver Wendt: Operations Research Script, Summer Term 2013
- Immanuel M. Bomze/W. Grossmann: Optimierung - Theorie und Algorithmen, ISBN:3-411-1509-1
- Kurt Marti/Detlef Gröger: Einführung in die lineare und nichtlineare Optimierung, ISBN:3-790-81297-8
- Wolfgang Domschke/Andreas Drexl: Einführung in Operations Research 6. Auflage ISBN:3-540-23431-4
- Hans Corsten/Hilde Corsten/Carsten Sartor: Operations Research ISBN:9-783800-632022
