MultiDimensionalSolver.DerivSteepestDescent Method 

Finds the location of a local extremum (i.e. minimum or maximum) of a differentiable multidimensional function using the method of Steepest Descent.

Parameters

extremumType

Indicates the type of extreme searched: in the case of a minimum, extremumType == ExtremumTypes.MINIMUM; and in the case of a maximum, extremumType == ExtremumTypes.MAXIMUM.

initialPoint

The initial point about which the iterative optimization procedure starts. The initial point is given as an array where the n-th term of the array corresponds to the value of the n coordinate value.

bracketingAlgorithm

The bracketing algorithm used within this method. We recommend in most cases that the AccelDerivBracketing bracketing algorithm is used. See the remarks given above, the PDF documentation or the examples for more information on selecting the most appropriate Bracketing Algorithm.

locateAlgorithm

The locate algorithm used within this method. We recommend in most cases that the CubicDerivLocate locate algorithms is used. See the remarks given above, the PDF documentation or the examples for more information on selecting the most appropriate Locate Algorithm.

minDistance

This parameter is required by all bracketing algorithms, and usually signifies the minimal distance between the end points of any bracketed interval in which the extremum lies. Therefore, the minDistance parameter should be approximately the same magnitude as tolerance parameter.

bracketingParameter

This parameter is required by some bracketing algorithms. For a detailed explanation as to its meaning and use for particular bracketing algorithms selected please see the documentation of the BracketingAlgorithm selected or the Mathematical Documentation chapter of the accompanying PDF documentation.

tolerance

The tolerance used when the exit condition of this iterative algorithm is checked. The smaller the tolerance used the more stable the convergence to the extremum within the iterative algorithm must be before the exit condition is trigger. Hence the smaller the tolerance generally the more accuracy the result, however it also has the draw back of making the algorithm more computationally demanding. A reasonable value to use for the tolerance is 0.1.

maxIterations

The maximum number of iterative steps taken before the algorithm exits and a TooManyMultiDimensionalIterationsException is thrown. A reasonable number to use for this parameter is 300.

fractionalTolerance

When the difference between the function values in two consecutive steps is less than fractionalTolerance, the point in the last step is considered a local extreme and returned. In most cases a reasonable value to take for this parameter is 0.0001, however if the object function in very oscillatory in a neighborhood of its extremum then a correspondingly larger value should be used.

Return Value

the point (an N-dimensional vector) where the function has a local extremum.

Remarks

Overview

This algorithm proceeds by taking a series of steps along the steepest downhill direction which we are able to evaluate because we are able to evaluate the functions gradient. After each steepest downhill step we imply a unidimensional minimization algorithm, which is determined by the bracketing algorithm (i.e. BracketingAlgorithm) and Locate Algorithm (i.e. LocateAlgorithm) chosen.

Remark: Since the steepest downhill steps are in a sense ideal, the locate algorithm used does not need to be particularly precise. Moreover, the precision of the Locate algorithm will only influence the speed at which the extremum is found and not the accuracy of the final result.

It is recommended that you use AccelDerivBracketing and CubicDerivLocate.

Exiting Conditions

After this algorithm has be run it will exit when one of the following two conditions has been satisfied:

1. Extremum found to the given level of tolerance set within the algorithm.
2. Maximum number of Iterations reached: If maxIterations is exceeded, then TooManyMultiDimensionalIterationsException is thrown resulting in the algorithm exiting.

Either way the algorithm is certain to exit. If the algorithm exits because the number of iterations has been exceeded then you have the option to re-run the algorithm with either a large number of iterations allowed and/or a higher level of tolerance allow.

Exceptions

See Also

MultiDimensionalSolver Class | WebCab.COM.Math.Optimization.MultiDimensional Namespace

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