PowellOptimizer Class

Use the PowellOptimizer class to find an extremum of a multivariate function using Powell's direction set method. The method is useful when the gradient of the objective function is not available. Still, this method is usually less efficient than either a quasi-Newton or a full conjugate gradient method with a numerical approximation of the gradient. It is here mostly for its historical importance.

The objective function must be supplied as a multivariate function delegate to the ObjectiveFunction property. The gradient of the objective function can be supplied either as a multivariate function returning a vector delegate (by setting the GradientFunction property), or a multivariate function returning a vector in its second argument delegate (by setting the FastGradientFunction property). The latter has the advantage that the same Vector instance is reused to hold the result.

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 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 change in value of the objective function at the approximate extremum. The test is successful when the change of the value of the objective function 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 Powell's method, and it's Active property is set to falseFalsefalsefalse (False in Visual Basic) before the algorithm is executed.