Use the NelderMeadOptimizer class to find an extremum of an objective function for which only the objective function is available, and the objective function itself may not be smooth. The method is often called the downhill simplex method.

The main advantage of this method is that it converges for functions where other methods would fail. This may happen, for instance, when the derivative of the objective function contains discontinuities. The major drawback is that the method converges more slowly than the other methods, and performs poorly for large problems.

The objective function must be supplied as a multivariate function delegate to the ObjectiveFunction property. The gradient of the objective function is not used.

Before the algorithm is run, you must set the InitialGuess property to a vector that contains an initial estimate for the extremum. The ExtremumType property specifies whether a minimum or a maximum of the objective function is desired.

The FindExtremum()()()() method performs the actual search for an extremum, and returns a Vector containing the best approximation. The Extremum property also returns the best approximation to the extremum. The ValueAtExtremum property returns the value of the objective function at the extremum.

The Status property is a AlgorithmStatus value that indicates the outcome of the algorithm. A value of Normal shows normal termination. A value of Divergent usually indicates that the objective function is not bounded.

The algorithm has two convergence tests. By default, the algorithm terminates when either of these is satisfied. You can deactivate either test by setting its Active property to falseFalsefalsefalse (False in Visual Basic). If both tests are deactivated, then the algorithm always terminates when the maximum number of iterations or function evaluations is reached.

The first test is based on the uncertainty in the location of the approximate extremum. The test succeeds if the difference between the best and worst approximation is within the tolerance. The SolutionTest property returns a VectorConvergenceTest object that allows you to specify the desired Tolerance and specific ConvergenceCriterion. See the VectorConvergenceTest class for details on how to further customize this test.

The second test is based on the difference in value of the objective function at the best and at the worst current approximation. The test is successful when the difference between the two is within the tolerance. Care should be taken with this test. When the tolerance is too large, the algorithm will terminate prematurely. The ValueTest property returns a SimpleConvergenceTest object that can be used to customize the test.

A third test, based on the value of the gradient at the approximate extremum, is exposed through the GradientTest property. However, this test does not apply to the Nelder-Mead method, and it's Active property is set to falseFalsefalsefalse (False in Visual Basic) before the algorithm is executed.

Numerical Libraries