Nonlinear Opt.: Basic concepts 2: Unterschied zwischen den Versionen
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<math>f'(x) = 2x - 8 ~~=> ~2x - 8 = 0 ~<=>~ x = 4</math> | <math>f'(x) = 2x - 8 ~~=> ~2x - 8 = 0 ~<=>~ x = 4</math> | ||
− | '''Step 2:''' Now you | + | '''Step 2:''' Now you have to 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> | <math>f''(x) = 2 ~> 0 ~~=> minimum </math> |
Version vom 22. Juni 2013, 16:23 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} \le \\ = \\ \ge \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 have to 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