com.numericalmethod.suanshu.optimization.unconstrained
Class NelderMead

java.lang.Object
  com.numericalmethod.suanshu.optimization.unconstrained.NelderMead

All Implemented Interfaces:

Minimizer, UnconstrainedMinimizer

public class NelderMead
extends java.lang.Object
implements UnconstrainedMinimizer

The Nelder–Mead method is a nonlinear optimization technique, which is well-defined for twice differentiable and unimodal problems. However, the Nelder–Mead technique is only a heuristic, since it can converge to non-stationary points on problems that can be solved by alternative methods.

The Nelder–Mead method starts with a simplex, and then uses operations reflection, expansion, contraction, and reduction, to search for the minimum/maximum point.

The Nelder–Mead algorithm does not always converge even for smooth problems. In practice, however, the performance is generally good.

Note that one dimensional optimization by Nelder-Mead is unreliable.

See Also:

Wikipedia: Nelder–Mead method

Nested Class Summary

static interface

NelderMead.BuildSimplex The Nelder–Mead algorithm requires an initial simplex to start the search.

Field Summary

double	alpha the reflection coefficient
double	gamma the expansion coefficient
double	rho the contraction coefficient
double	sigma the shrink/reduction coefficient

Constructor Summary

NelderMead()
Construct a NelderMead instance to minimize an objective function with the default control parameters.

NelderMead(double alpha, double gamma, double rho, double sigma)
Construct a NelderMead instance to minimize/maximize f with the control parameters.

Method Summary

double	minimum() Get the minimum found.
double[]	search(double[] initial) Perform a Nelder-Mead search from one initial guess.
double[]	search(int maxIterations, double[]... simplex) Perform a Nelder-Mead search from an initial simplex.
double[]	search(int maxIterations, double[] initial) Perform a Nelder-Mead search from one initial guess.
Vector	search(int maxIterations, Vector... simplex) Perform a Nelder-Mead search from an initial simplex.
Vector	search(int maxIterations, Vector initial, NelderMead.BuildSimplex simplex) Perform a Nelder-Mead search from one initial guess.
void	solve(RealScalarFunction f, double tol)
void	solve(UnconstrainedProblem problem, double tol)

Methods inherited from class java.lang.Object

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

Field Detail

alpha

public final double alpha

the reflection coefficient

gamma

public final double gamma

the expansion coefficient

rho

public final double rho

the contraction coefficient

sigma

public final double sigma

the shrink/reduction coefficient

Constructor Detail

NelderMead

public NelderMead(double alpha,
                  double gamma,
                  double rho,
                  double sigma)

Construct a NelderMead instance to minimize/maximize f with the control parameters.

Parameters:

alpha - the reflection coefficient

gamma - the shrink/reduction coefficient

rho - the contraction coefficient

sigma - the shrink/reduction coefficient

NelderMead

public NelderMead()

Construct a NelderMead instance to minimize an objective function with the default control parameters.

Method Detail

solve

public void solve(UnconstrainedProblem problem,
                  double tol)

Specified by:

solve in interface UnconstrainedMinimizer

solve

public void solve(RealScalarFunction f,
                  double tol)

Specified by:

solve in interface UnconstrainedMinimizer

minimum

public double minimum()

Description copied from interface: Minimizer

Get the minimum found.

This is especially useful for an objective function that takes a long time to compute.

Specified by:

minimum in interface Minimizer

Returns:

the minimum found

search

public double[] search(int maxIterations,
                       double[]... simplex)

Perform a Nelder-Mead search from an initial simplex.

Parameters:

maxIterations - the maximum number of iterations

simplex - initial guesses for the optimizer. The guess points should form a simplex.

Returns:

the optimal point

search

public Vector search(int maxIterations,
                     Vector initial,
                     NelderMead.BuildSimplex simplex)

Perform a Nelder-Mead search from one initial guess.

Parameters:

maxIterations - the maximum number of iterations

initial - an initial guess of the optimizer

simplex - specify the way to build a simplex from an initial point

Returns:

the optimal point

search

public double[] search(int maxIterations,
                       double[] initial)

Perform a Nelder-Mead search from one initial guess.

Parameters:

maxIterations - the maximum number of iterations

initial - an initial guess of the optimizer

Returns:

the optimal point

search

public double[] search(double[] initial)

Perform a Nelder-Mead search from one initial guess.

Parameters:

initial - an initial guess of the optimizer

Returns:

the optimal point

search

public Vector search(int maxIterations,
                     Vector... simplex)

Perform a Nelder-Mead search from an initial simplex.

Specified by:

search in interface Minimizer

Parameters:

maxIterations - the maximum number of iterations, e.g., 500

simplex - initial guesses for the optimizer; The guess points should form a simplex.

Returns:

the optimal point