NLopt Tutorial

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==Example nonlinearly constrained problem== ==Example nonlinearly constrained problem==
- +[[Image:NLopt-example-constraints.png|right|thumb|400px|Feasible region for a simple example optimization problem with two nonlinear (cubic) constraints.]]
As a first example, we'll look at the nonlinearly constrained minimization problem: As a first example, we'll look at the nonlinearly constrained minimization problem:
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for parameters ''a''<sub>1</sub>=2, ''b''<sub>1</sub>=0, ''a''<sub>2</sub>=-1, ''b''<sub>2</sub>=1. for parameters ''a''<sub>1</sub>=2, ''b''<sub>1</sub>=0, ''a''<sub>2</sub>=-1, ''b''<sub>2</sub>=1.
-[[Image:NLopt-example-constraints.png|right|thumb|400px|Feasible region for a simple example optimization problem with two nonlinear (cubic) constraints.]]+ 
The feasible region defined by these constraints is plotted at right: ''x''<sub>2</sub> is constrained to lie at the maximum of two cubics, and the optimum point is located at the intersection (1/3, 8/27) where the objective function takes on the value <math>\sqrt{8/27} \approx 0.5443310539518\ldots</math>. The feasible region defined by these constraints is plotted at right: ''x''<sub>2</sub> is constrained to lie at the maximum of two cubics, and the optimum point is located at the intersection (1/3, 8/27) where the objective function takes on the value <math>\sqrt{8/27} \approx 0.5443310539518\ldots</math>.
(This problem is especially trivial, because by formulating it in terms of the cube root of ''x''<sub>2</sub> you can turn it into a linear-programming problem, but we won't do that here.) (This problem is especially trivial, because by formulating it in terms of the cube root of ''x''<sub>2</sub> you can turn it into a linear-programming problem, but we won't do that here.)

Revision as of 22:11, 13 November 2008

In this tutorial, we illustrate the usage of NLopt in various languages via one or two trivial examples.

Example nonlinearly constrained problem

Feasible region for a simple example optimization problem with two nonlinear (cubic) constraints.
Enlarge
Feasible region for a simple example optimization problem with two nonlinear (cubic) constraints.

As a first example, we'll look at the nonlinearly constrained minimization problem:

\min_{\mathbf{x}\in\mathbb{R}^2} \sqrt{x_2}
subject to x_2 \geq 0, x_2 \geq (a_1 x_1 + b_1)^3, and x_2 \geq (a_2 x_1 + b_2)^3

for parameters a1=2, b1=0, a2=-1, b2=1.


The feasible region defined by these constraints is plotted at right: x2 is constrained to lie at the maximum of two cubics, and the optimum point is located at the intersection (1/3, 8/27) where the objective function takes on the value \sqrt{8/27} \approx 0.5443310539518\ldots.

(This problem is especially trivial, because by formulating it in terms of the cube root of x2 you can turn it into a linear-programming problem, but we won't do that here.)

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