An implementation of the Nelder-Mead simplex method.
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An implementation of the Nelder-Mead simplex method.
Version 1.1

Code Example

See rosenbrock and maxpower.

The rosenbrock example finds the minimum for the Rosenbrock function.

In the first case a starting point of (-1.2,1.0) was specified.

alt text

Initial Values
-1.20, 1.00, value 24.20
-0.23, 1.26, value 147.22
-0.94, 1.97, value 121.79
123 Function Evaluations
61 Iterations through program
The minimum was found at
1.00, 1.00, value 0.00

In the second case no starting point was specified.

Initial Values
0.00, 0.00, value 1.00
0.97, 0.26, value 46.36
0.26, 0.97, value 81.98
102 Function Evaluations
50 Iterations through program
The minimum was found at
1.00, 1.00, value 0.00

The maxpower program is an example of using Nelder-Mead for a one dimensional optimization problem. The problem is to find the value of a load resistor that will allow maximum power to be delivered to the load. For this example I used a simple voltage divider circuit consisting of a single 9V source with two resistors in series (1000 ohm and 470 ohm) with the output taken across the 470 ohm resistor. From the Thevenin equivalent circuit we know that the optimium load resistor value for max power transfer should be 319.73 ohms. The source code for this problem can be found here. The program produces the following output starting with an initial guess of RL=100 ohms:

Initial Values
100.000000, value -0.004700
101.000000, value -0.004725
29 Function Evaluations
13 Iterations through program
320.00 ohms, 0.006474 watts


The Nelder-Mead Simplex Method is a direct search algorithm that's useful for non-linear optimization problems. I was researching optimization of antenna arrarys at one point and implemented several versions of the Nelder-Mead algorithm.

Reference for the creation of the initial simplex.
D. J. Wilde and C. S. Beightler, Foundations of Optimization. Englewood Cliffs, N.J., Prentice-Hall, 1967, p. 319

The pn,qn values used to create the initial simplex are defined by (Spendly, Hext, and Himsworth). This creates a simplex with unit edges between all vertices. Some implementations just form the initial simplex by taking a step in each direction. Since we're talking about non-linear optimization, there is no one initial simplex that is best for every case. For my applications, concerning antenna array optimization, the initial simplex with unit edges seems to give the best results. Although it does have to be scaled at times depending on the problem.




Copyright (c) 1997 Michael F. Hutt, Released under the MIT License