Nonlinear Opt.: Examples and Modeling 2

Nonlinear Optimization: Examples and Modeling

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Theory

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In mathematics, computer science, or management science, mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives. In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain, including a variety of different types of objective functions and different types of domains. As far as the restrictions, constrains and the objective function is concerned, the nonlinear optimization is equal to the linear optimization. The main difference between NLP and LP is that NLP can not be solved with the help of the simplex algorithm because there is a nonlinear objective function and/or at least one nonlinear restriction.

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Linear objective function and nonlinear restriction:

We have two products. Apples ($x_{1}$) and pears ($x_{2}$). Our goal is to maximize the marginal return of the two products. With the Gutenberg production function, we get our nonlinear restrictions. Together with the linear objective function we have the following exercise:

$price(x_{1})=6$

$price(x_{2})=4$

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): R = 6x_1+4x_2 \rightarrow max!

because of the bad weather and a lack of space, we have the following restrictions:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 5,5x_1^2+7,5x_2^2\leq45000

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 2,3x_1^2+11,2x_2^2\leq50000

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1, x_2 \geq 0

Nonlinear objective function and nonlinear restriction:

The goal is to produce a can which consists of a bottom, a lid and a lateral surface area. We want to be able to fit 1,5 litre in the can and use as less material as possible.

$(2r)^{2}$: Bottom/Lid

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 2\pi r h

lateral surface area

Objective function:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 2(2r)^2+2\pi r h\rightarrow min!

Restriction

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): V=\pi r^2 h = 1,5

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): r,h\geq 0

Nonlinear objective function and linear restriction:

There is a performance in the Pfalztheater in Kaiserslautern. The host wants it to be sold out(200 Seats). Because of the supervision, he wants that there is always one adult($x_{1}$) going to the performance per two children($x_{2}$). The entry fee is 10Euro for adults and only 5Euro for children. He wants to maximize his return.

Objective function

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 10x_1+3x_2^2\rightarrow max!

Restriction

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1+x_2\leq200

$x_{1}=2x_{2}$

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_1,x_2\geq 0

Sources

http://www.cs.hs-rm.de/~weber/or/weber/orfolien5.pdf

Kathöfer,U. Operation Research 2.Auflage 2008

Werners,B. Grundlagen des Operation Research 2008

Authors

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