com.imsl.math
Class MinUnconMultiVar

java.lang.Object
  com.imsl.math.MinUnconMultiVar

All Implemented Interfaces:

Serializable, Cloneable

public class MinUnconMultiVar
extends Object
implements Serializable, Cloneable

Unconstrained multivariate minimization.

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:

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

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

such that

Finally, the optimality condition where is a gradient tolerance, is sought.

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

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 gradientTolerance. The second stopping criterion for MinUnconMultivar occurs when the scaled distance between the last two steps is less than the step tolerance stepTolerance.

Since by default, a finite-difference method is used to estimate the gradient for some single precision calculations, 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 (setGradient).

The user may also choose to supply the Hessian. If, during intermediate calculations, the user supplied Hessian becomes non-symmetric positive definite, then the algorithm will resort to using the default approximate Hessian.

See Also:

Finite-difference Example, Gradient Example, Hessian Example, Serialized Form

Nested Class Summary

static class	MinUnconMultiVar.ApproximateMinimumException Scaled step tolerance satisfied; the current point may be an approximate local solution, or the algorithm is making very slow progress and is not near a solution, or the scaled step tolerance is too big.
static class	MinUnconMultiVar.FalseConvergenceException False convergence error; the iterates appear to be converging to a noncritical point.
static interface	MinUnconMultiVar.Function Public interface for the user supplied function to the MinUnconMultiVar object.
static interface	MinUnconMultiVar.Gradient Public interface for the user supplied gradient to the MinUnconMultiVar object.
static interface	MinUnconMultiVar.Hessian Public interface for the user supplied Hessian to the MinUnconMultiVar object.
static class	MinUnconMultiVar.MaxIterationsException Maximum number of iterations exceeded.
static class	MinUnconMultiVar.UnboundedBelowException Five consecutive steps of the maximum allowable stepsize have been taken, either the function is unbounded below, or has a finite asymptote in some direction or the maximum allowable step size is too small.

Constructor Summary

MinUnconMultiVar(int n)
Unconstrained minimum constructor for a function of n variables of type double.

Method Summary

double[]	computeMin(MinUnconMultiVar.Function F) Return the minimum point of a function of n variables of type double using a finite-difference gradient or using a user-supplied gradient.
int	getErrorStatus() Returns the non-fatal error status.
int	getIterations() Returns the number of iterations used to compute a minimum.
void	setDigits(double fdigit) Set the number of good digits in the function.
void	setFscale(double fscale) Set the function scaling value for scaling the gradient.
void	setGradientTolerance(double gradientTolerance) Sets the gradient tolerance.
void	setGuess(double[] xguess) Set the initial guess of the minimum point of the input function.
void	setIhess(int ihess) Set the Hessian initialization parameter.
void	setMaximumStepsize(double maximumStepsize) Set the maximum allowable stepsize to use.
void	setMaxIterations(int maxIterations) Set the maximum number of iterations allowed.
void	setStepTolerance(double stepTolerance) Set the scaled step tolerance to use when changing x.
void	setXscale(double[] xscale) Set the diagonal scaling matrix for the variables.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

MinUnconMultiVar

public MinUnconMultiVar(int n)

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

Parameters:

n - An int scalar value which defines the number of variables of the function whose minimum is to be found.

Method Detail

computeMin

public double[] computeMin(MinUnconMultiVar.Function F)
                    throws MinUnconMultiVar.FalseConvergenceException,
                           MinUnconMultiVar.MaxIterationsException,
                           MinUnconMultiVar.UnboundedBelowException

Return the minimum point of a function of n variables of type double using a finite-difference gradient or using a user-supplied gradient.

Parameters:

F - defines the function whose minimum is to be found. F can be used to supply a gradient of the function. If F implements Gradient then the user-supplied gradient is used. Otherwise, an attempt to find the minimum is made using a finite-difference gradient. If F implements Hessian then the user-supplied gradient and Hessian are used. Otherwise, an approximate Hessian is used during computation.

Returns:

a double array containing the point at which the minimum of the input function occurs.

Throws:

MinUnconMultiVar.FalseConvergenceException

MinUnconMultiVar.MaxIterationsException

MinUnconMultiVar.UnboundedBelowException

getErrorStatus

public int getErrorStatus()

Returns the non-fatal error status.

Returns:

an int specifying the non-fatal error status:

Status	Meaning
1	The last global step failed to locate a lower point than the current x value. The current x may be an approximate local minimizer and no more accuracy is possible or the step tolerance may be too large.
2	Relative function convergence; both the actual and predicted relative reductions in the function are less than or equal to the relative function convergence tolerance.
3	Scaled step tolerance satisfied; the current point may be an approximate local solution, or the algorithm is making very slow progress and is not near a solution, or the step tolerance is too big.

getIterations

public int getIterations()

Returns the number of iterations used to compute a minimum.

Returns:

an int specifying the number of iterations used to compute the minimum.

setDigits

public void setDigits(double fdigit)

Set the number of good digits in the function. If this member function is not called, fdigit is set to 15.0.

Parameters:

fdigit - a double scalar value specifying the number of good digits in the user supplied function

Throws:

IllegalArgumentException - is thrown if fdigit is less than or equal to 0

setFscale

public void setFscale(double fscale)

Set the function scaling value for scaling the gradient. If this member function is not called, the value of this scalar is set to 1.0.

Parameters:

fscale - a double scalar specifying the function scaling value for scaling the gradient

Throws:

IllegalArgumentException - is thrown if fscale is less than or equal to 0.

setGradientTolerance

public void setGradientTolerance(double gradientTolerance)

Sets the gradient tolerance. This first stopping criterion for this optimizer is that the norm of the gradient be less than the gradient tolerance. If this member function is not called, the cube root of machine precision squared is used to compute the gradient.

Parameters:

gradientTolerance - a double specifying the gradient tolerance used to compute the gradient

Throws:

IllegalArgumentException - is thrown if gradientTolerance is less than or equal to 0

setGuess

public void setGuess(double[] xguess)

Set the initial guess of the minimum point of the input function. If this member function is not called, the elements of this array are set to 0.0..

Parameters:

xguess - a double array specifying the initial guess of the minimum point of the input function

setIhess

public void setIhess(int ihess)

Set the Hessian initialization parameter. If this member function is not called, ihess is set to 0.0 and the Hessian is initialized to the identity matrix. If this member function is called and ihess is set to anything other than 0.0, the Hessian is initialized to the diagonal matrix containing max(abs(f(xguess)),fscale)*xscale*xscale

Parameters:

ihess - an int scalar value specifying the Hessian initialization parameter. If ihess = 0.0 the Hessian is initialized to the identity matrix. Otherwise, the Hessian is initialized to the diagonal matrix containing max(abs(f(xguess)),fscale)*xscale*xscale where xguess is the initial guess of the computed solution and xscale is the scaling vector for the variables.

setMaximumStepsize

public void setMaximumStepsize(double maximumStepsize)

Set the maximum allowable stepsize to use. If this member function is not called, maximum stepsize is set to a default value based on a scaled xguess.

Parameters:

maximumStepsize - a nonnegative double value specifying the maximum allowable stepsize

Throws:

IllegalArgumentException - is thrown if maximumStepsize is less than or equal to 0

setMaxIterations

public void setMaxIterations(int maxIterations)

Set the maximum number of iterations allowed. If this member function is not called, the maximum number of iterations is set to 100.

Parameters:

maxIterations - an int specifying the maximum number of iterations allowed

Throws:

IllegalArgumentException - is thrown if maxIterations is less than or equal to 0

setStepTolerance

public void setStepTolerance(double stepTolerance)

Set the scaled step tolerance to use when changing x. If this member function is not called, the scaled step tolerance is set to 3.66685e-11.

The second stopping criterion for this optimizer is that the scaled distance between the last two steps be less than the step tolerance.

Parameters:

stepTolerance - a double scalar value specifying the scaled step tolerance. The i-th component of the scaled step between two points x and y is computed as abs(x(i)-y(i))/max(abs(x(i)),1/xscale(i)) where xscale is the scaling vector for the variables.

Throws:

IllegalArgumentException - is thrown if stepTolerance is less than or equal to 0

setXscale

public void setXscale(double[] xscale)

Set the diagonal scaling matrix for the variables. If this member function is not called, the elements of this array are set to 1.0..

Parameters:

xscale - a double array specifying the diagonal scaling matrix for the variables

Throws:

IllegalArgumentException - is thrown if any of the elements of xscale is less than or equal to 0