# LimitedMemoryBfgsOptimizer Class

Represents a multi-dimensional optimizer that uses the limited memory variant of the BFGS algorithm.

## Definition

Namespace: Extreme.Mathematics.Optimization
Assembly: Extreme.Numerics (in Extreme.Numerics.dll) Version: 8.1.23
C#
``public sealed class LimitedMemoryBfgsOptimizer : DirectionalOptimizer``
Inheritance
Object  →  ManagedIterativeAlgorithm<Vector<Double>, Double, OptimizationSolutionReport>  →  MultidimensionalOptimizer  →  DirectionalOptimizer  →  LimitedMemoryBfgsOptimizer

## Remarks

Use the LimitedMemoryBfgsOptimizer class to find an extremum of a multivariate function using a quasi-Newton method where the number of variables is relatively large. This is the preferred method for larger problems when the gradient of the objective function is available. An alternative is the Conjugate Gradient method.

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.

Sometimes, the gradient function is not available, or is very expensive to calculate. In such instances, a numerical approximation may work better.

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 three convergence tests. By default, the algorithm terminates when either of these is satisfied. You can deactivate either test by setting its Enabled property to false. If both tests are deactivated, { 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<T> object that allows you to specify the desired Tolerance and specific ConvergenceCriterion. See the VectorConvergenceTest<T> 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<T> 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<T> object that can be used to customize the test. By default, the error is set to the component with the largest absolute value.

## Constructors

 LimitedMemoryBfgsOptimizer() Constructs a new QuasiNewtonOptimizer object. LimitedMemoryBfgsOptimizer(Int32) Constructs a new LimitedMemoryBfgsOptimizer object.