BoundedLeastSquares Class

Solves a nonlinear least-squares problem subject to bounds on the variables using a modified Levenberg-Marquardt algorithm.

Inheritance Hierarchy

SystemObject
Imsl.MathBoundedLeastSquares

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class BoundedLeastSquares

<SerializableAttribute>
Public Class BoundedLeastSquares

[SerializableAttribute]
public ref class BoundedLeastSquares

[<SerializableAttribute>]
type BoundedLeastSquares =  class end

The BoundedLeastSquares type exposes the following members.

Constructors

	Name	Description
	BoundedLeastSquares	Constructor for BoundedLeastSquares.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetJacobianSolution	Returns the Jacobian at the approximate solution.
	GetResiduals	Returns the residuals at the approximate solution.
	GetSolution	Returns the solution.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetFscale	Sets the diagonal scaling matrix for the functions.
	SetGuess	Sets the initial guess of the solution.
	SetInternalScale	The internal variable scaling option.
	SetXscale	The scaling vector for the variables.
	Solve	Solves a nonlinear least-squares problem subject to bounds on the variables using a modified Levenberg-Marquardt algorithm.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	AbsoluteTolerance	The absolute function tolerance.
	Digits	The number of good digits in the function.
	GradientTolerance	The scaled gradient tolerance.
	MaximumFunctionEvals	The maximum number of function evaluations.
	MaximumIterations	The maximum number of iterations.
	MaximumJacobianEvals	The maximum number of Jacobian evaluations.
	MaximumStepsize	The maximum allowable step size.
	NumberOfProcessors	Perform the parallel calculations with the maximum possible number of processors set to NumberOfProcessors.
	Parallel	Enable or disable performing MinUnconMultiVar.IFunction.F in parallel.
	RelativeTolerance	The relative function tolerance.
	ScaledStepTolerance	The scaled step tolerance.
	TrustRegion	The size of initial trust region radius.

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Remarks

Class BoundedLeastSquares uses a modified Levenberg-Marquardt method and an active set strategy to solve nonlinear least-squares problems subject to simple bounds on the variables. The problem is stated as follows:

$min {1 \over 2}F\left( x \right)^T F\left( x \right) = {1 \over 2}\sum\limits_{i = 1}^m {f_i } \left( x \right)^2$

subject to

$\,l \le x \le u$

here ${\rm{m }} \ge {\rm{ n}}$ , ${\it F}: {\rm{R}}^n \to {\rm{R}}^m$ , and

is the i-th component function of F(x). From a given starting point, an active set IA, which contains the indices of the variables at their bounds, is built. A variable is called a "free variable" if it is not in the active set. The routine then computes the search direction for the free variables according to the formula

$d = - \left( {J^T J + \mu I} \right)^{ - 1} J^T F$

where $\mu$ is the Levenberg-Marquardt parameter, F = F(x), and J is the Jacobian with respect to the free variables. The search direction for the variables in IA is set to zero. The trust region approach discussed by Dennis and Schnabel (1983) is used to find the new point. Finally, the optimality conditions are checked. The conditions are:

$\left\| {g\left( {x_i{} } \right)} \right\| \le \varepsilon ,l_i \lt x_i \lt u_i$

$g\left( {x_i } \right) \lt 0,x_i = u_i$

$g\left( {x_i } \right) \gt 0,x_i = l_i$

where $\varepsilon$ is a gradient tolerance. This process is repeated until the optimality criterion is achieved.

The active set is changed only when a free variable hits its bounds during an iteration or the optimality condition is met for the free variables but not for all variables in IA, the active set. In the latter case, a variable that violates the optimality condition will be dropped out of IA. For more details on the Levenberg-Marquardt method, see Levenberg (1944) or Marquardt (1963). For more detail on the active set strategy, see Gill and Murray (1976).

Reference

Imsl.Math Namespace

Other Resources

Example 1

Example 2