Nonlinear Opt.: Basic concepts 2: Unterschied zwischen den Versionen

Version vom 22. Juni 2013, 18:29 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.

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.

${\displaystyle Objective:Max~(orMin)z=F(x)}$

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): Restrictions: g_i(x) \begin{Bmatrix} \le \\ = \\ \ge \end{Bmatrix}~0 ~~~~~~(for~ i ~= ~1,...,m)

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x \in W_1 \times W_2 \times ,..., \times W_n , W_j \in \{ \mathbb R_+, \mathbb Z_+, \mathbb B \} ~~~~~~~~~~(for~j = 1,...,n)

In contrast to linear problems, non-linear optimization problems got a non-linear objective function and/or at least one non-linear restriction. As distinct from integer and combinatorial problems, where integer or binary variables occur, we assume that only (non-negative) real variables occur in non-linear optimization. The to be minimized objective function can be replaced with the to be maximized objective function and vice versa. Furthermore, every Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \ge

– restriction can be transformed into a Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \le
– testriction by multiplication with -1. Beyond that, all equations can be replaced by two Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \le
– restrictions.

To simplify the notation, functions ${\displaystyle g_{i}}$ consist of the constants ${\displaystyle b_{i}}$, which are designated within the linear optimization:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): f_i(x) \le b_i

is transformed into

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): g_i(x) := f_i (x)-b_i \le 0

The following notation is assumed to be the notation of a non-linear optimization problem:

maximize F(x) under the restrictions

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): g_i (x) \le 0 ~~for~~ i = 1,...,m

and

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_j \ge 0 ~~for~~ j = 1,...,n

Examples

The first two examples are very easy, because there are not any restrictions. Example 1 is a simple minimization problem and example 2 a maximization problem.

Example 1

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.

${\displaystyle f''(x)=2~>0~~=>minimum}$

Example 2

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.

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): f''(x) = -2 ~< 0 ~~=> maximum

How to solve different types of non-linear problems

That is not a basic concept and therefore explained in other wiki-entries.

Class 1 (non-linear objective function, no restrictions)

• Classical gradient method
• Golden section search

Class 2 (non-linear objective function, linear restrictions)

• Lagrangian condition
• Karush-Kuhn-Tucker

Class 3 (non-linear objective function, non-linear restrictions)

• Lagrangian condition
• Karush-Kuhn-Tucker

Class 4 (linear objective function, non-linear restrictions)

Sources

Internet sources

• http://de.wikipedia.org/wiki/Optimierung_(Mathematik)#Nichtlineare_Optimierung

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