MinUnconMultiVar Class

Minimizes a multivariate function using a quasi-Newton method.

Inheritance Hierarchy

SystemObject
Imsl.MathMinUnconMultiVar

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

Syntax

C++

Copy

[SerializableAttribute]
public class MinUnconMultiVar

<SerializableAttribute>
Public Class MinUnconMultiVar

[SerializableAttribute]
public ref class MinUnconMultiVar

[<SerializableAttribute>]
type MinUnconMultiVar =  class end

The MinUnconMultiVar type exposes the following members.

Constructors

	Name	Description
	MinUnconMultiVar	Unconstrained minimum constructor for a function of n variables of type double.

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Methods

	Name	Description
	ComputeMin	Return the minimum point of a function of n variables of type double using a finite-difference gradient or using a user-supplied gradient.
	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.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetGuess	Sets the initial guess of the minimum point of the input function.
	SetXscale	Sets the diagonal scaling matrix for the variables.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	Digits	The number of good digits in the function.
	ErrorStatus	The non-fatal error status.
	Fscale	The function scaling value for scaling the gradient.
	GradientTolerance	The gradient tolerance used to compute the gradient.
	Ihess	The Hessian initialization parameter.
	Iterations	The number of iterations used to compute a minimum.
	MaximumStepsize	The maximum allowable stepsize to use.
	MaxIterations	The maximum number of iterations allowed.
	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.
	StepTolerance	The scaled step tolerance to use when changing x.

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Remarks

Class MinUnconMultivar uses a quasi-Newton method to find the minimum of a function f(x) of n variables. The problem is stated as follows:

$\mathop {\min }\limits_{x\; \in \;R^n } f\left ( x \right)$

Given a starting point , the search direction is computed according to the formula

$d = -B^{-1} g_c$

where B is a positive definite approximation of the Hessian, and is the gradient evaluated at . A line search is then used to find a new point

$x_n = \,x_c + \lambda d,\lambda > 0$

such that

$f\left( {x_n } \right) \le f\left( {x_c } \right) + \alpha g^T d,\,\,\,\, \alpha \in \left( {0,\,0.5} \right)$

Finally, the optimality condition ${\rm{||g(x)|| }} \le \varepsilon$ where $\varepsilon$ is a gradient tolerance.

When optimality is not achieved, B is updated according to the BFGS formula

$B \leftarrow B - \frac{{Bss^T B}}{{s^T Bs}} + \frac{{yy^T }}{{y^T s}}$

where and . Another search direction is then computed to begin the next iteration. For more details, see Dennis and Schnabel (1983, Appendix A).

In this implementation, the first stopping criterion for MinUnconMultivar occurs when the norm of the gradient is less than the given gradient tolerance property, GradientTolerance. The second stopping criterion for MinUnconMultivar occurs when the scaled distance between the last two steps is less than the step tolerance property, StepTolerance.

Since by default, a finite-difference method is used to estimate the gradient. An inaccurate estimate of the gradient may cause the algorithm to terminate at a noncritical point. Supply the gradient for a more accurate gradient evaluation (MinConMultiVar.IGradient).

Reference

Imsl.Math Namespace

Other Resources

Finite-difference Example

Gradient Example