ConjugateGradientOptimizer Class

Use the ConjugateGradientOptimizer class to solve an optimization problem using a conjugate gradient algorithm. Three variants of the algorithm are available: the method of Fletcher and Reeves, the method of Polak and Ribière, and the positive method of Polak and Ribière. The default is the positive Polak-Ribière method.

The conjugate gradient method is the method of choice for large problems. For these problems, it consumes less memory and performs less work per iteration than the other common methods. On the downside, the search directions are often badly scaled, which makes the method less suitable for smaller problems. A quasi-Newton algorithm is preferred in such a case.

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.

A number of properties let you control the search for an extremum. The LineSearch property returns a OneDimensionalOptimizer that is used to locate a suitable new point along the current search direction. You can modify its convergence criteria. Note that conjugate gradient algorithms require a fairly precise line search.

Sometimes, successive conjugate directions are almost parallel, or don't reflect the current curvature of the objective function well, resulting in poor convergence. This can be remedied in one of two ways. The RestartIterations property specifies how often the conjugate direction is to be reset to the steepest descent direction. A value of 0, which is the default, specifies not to reset the direction. The RestartThreshold property is used to test whether successive search directions are sufficiently orthogonal. If this is the case, then the search direction is reset to the steepest descent direction. A lower value indicates more frequent resets. The default value is 0.1.

The algorithm has three 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.

The third test is based on the value of the gradient at the approximate extremum. The GradientTest property returns a VectorConvergenceTest object that can be used to customize the test. By default, the error is set to the component with the largest absolute value.

System..::..Object
  Extreme.Mathematics.Algorithms..::..ManagedIterativeAlgorithm<(Of <(<'Vector>)>)>
    Extreme.Mathematics.Optimization..::..MultidimensionalOptimizer
      Extreme.Mathematics.Optimization..::..DirectionalOptimizer
        Extreme.Mathematics.Optimization..::..ConjugateGradientOptimizer