NelderMead Class

Nelder-Mead simplex algorithm with support for bound constraints for non-linear, gradient-free optimization.

Inheritance Hierarchy

SystemObject
Accord.Math.OptimizationBaseOptimizationMethod
Accord.Math.OptimizationNelderMead

Namespace: Accord.Math.Optimization
Assembly: Accord.Math (in Accord.Math.dll) Version: 3.8.0

Syntax

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public class NelderMead : BaseOptimizationMethod, IOptimizationMethod<NelderMeadStatus>, 
	IOptimizationMethod, IOptimizationMethod<double[], double>, IOptimizationMethod<double[], double, NelderMeadStatus>

Public Class NelderMead
	Inherits BaseOptimizationMethod
	Implements IOptimizationMethod(Of NelderMeadStatus), IOptimizationMethod, 
	IOptimizationMethod(Of Double(), Double), IOptimizationMethod(Of Double(), Double, NelderMeadStatus)

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The NelderMead type exposes the following members.

Constructors

	Name	Description
	NelderMead(Int32)	Creates a new NelderMead non-linear optimization algorithm.
	NelderMead(NonlinearObjectiveFunction)	Creates a new NelderMead non-linear optimization algorithm.
	NelderMead(Int32, FuncDouble, Double)	Creates a new NelderMead non-linear optimization algorithm.

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Properties

	Name	Description
	Capacity	Gets the maximum number of variables that can be optimized by this instance. This is the initial value that has been passed to this class constructor at the time the algorithm was created.
	Convergence	Gets or sets multiple convergence options to determine when the optimization can terminate.
	DiameterTolerance	Gets or sets the by how much the simplex diameter \|xl - xh\| must be reduced before the algorithm can be terminated. Setting this value to a value higher than zero causes the algorithm to replace the standard Convergence criteria with this condition. Default is zero.
	Difference	The difference between the high and low function values of the last simplex in the previous call to the optimization function.
	Function	Gets or sets the function to be optimized. (Inherited from BaseOptimizationMethod.)
	LowerBounds	Gets the lower bounds that should be respected in this optimization problem. Default is to initialize this vector with NegativeInfinity.
	MaximumValue	Gets or sets the maximum value that the objective function could produce before the algorithm could be terminated as if the solution was good enough.
	NumberOfVariables	Gets or sets the number of variables (free parameters) in the optimization problem. This number can be decreased after the algorithm has been created so it can operate on subspaces. (Overrides BaseOptimizationMethodNumberOfVariables.)
	Solution	Gets the current solution found, the values of the parameters which optimizes the function. (Inherited from BaseOptimizationMethod.)
	Status	Get the exit code returned in the last call to the Maximize or Minimize methods.
	StepSize	Gets the step sizes to be used by the optimization algorithm. Default is to initialize each with 1e-5.
	Token	Gets or sets a cancellation token that can be used to stop the learning algorithm while it is running. (Inherited from BaseOptimizationMethod.)
	UpperBounds	Gets the upper bounds that should be respected in this optimization problem. Default is to initialize this vector with PositiveInfinity.
	Value	Gets the output of the function at the current Solution. (Inherited from BaseOptimizationMethod.)

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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 the default hash function. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	Maximize	Finds the maximum value of a function. The solution vector will be made available at the Solution property. (Inherited from BaseOptimizationMethod.)
	Maximize(Double)	Finds the maximum value of a function. The solution vector will be made available at the Solution property. (Inherited from BaseOptimizationMethod.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	Minimize	Finds the minimum value of a function. The solution vector will be made available at the Solution property. (Inherited from BaseOptimizationMethod.)
	Minimize(Double)	Finds the minimum value of a function. The solution vector will be made available at the Solution property. (Inherited from BaseOptimizationMethod.)
	Minimize(Double)	Finds the minimum value of a function, using the function output at the current value, if already known. This overload can be used when embedding Nelder-Mead in other algorithms to avoid initial checks.
	OnNumberOfVariablesChanged	Called when the NumberOfVariables property has changed. (Overrides BaseOptimizationMethodOnNumberOfVariablesChanged(Int32).)
	Optimize	Implements the actual optimization algorithm. This method should try to minimize the objective function. (Overrides BaseOptimizationMethodOptimize.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Extension Methods

	Name	Description
	HasMethod	Checks whether an object implements a method with the given name. (Defined by ExtensionMethods.)
	IsEqual	Compares two objects for equality, performing an elementwise comparison if the elements are vectors or matrices. (Defined by Matrix.)
	To(Type)	Overloaded. Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime. (Defined by ExtensionMethods.)
	ToT	Overloaded. Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime. (Defined by ExtensionMethods.)

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Remarks

The Nelder–Mead method or downhill simplex method or amoeba method is a commonly used nonlinear optimization technique, which is a well-defined numerical method for problems for which derivatives may not be known. However, the Nelder–Mead technique is a heuristic search method that can converge to non-stationary points on problems that can be solved by alternative methods.

The Nelder–Mead technique was proposed by John Nelder and Roger Mead (1965) and is a technique for minimizing an objective function in a many-dimensional space.

The source code presented in this file has been adapted from the Sbplx method (based on Nelder-Mead's Simplex) given in the NLopt Numerical Optimization Library, created by Steven G. Johnson.

References:

Examples

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// Let's say we would like to find the minimum 
// of the function "f(x) = 10 * (x+1)^2 + y^2".

// In code, this means we would like to minimize:
Func<double[], double> function = (double[] x) =>
    10.0 * Math.Pow(x[0] + 1.0, 2.0) + Math.Pow(x[1], 2.0);

// We can do so using the NelderMead class:
var solver = new NelderMead(numberOfVariables: 2)
{
    Function = function // f(x) = 10 * (x+1)^2 + y^2
};

// Now, we can minimize it with:
bool success = solver.Minimize();

// And get the solution vector using
double[] solution = solver.Solution; // should be (-1, 1)

// The minimum at this location would be:
double minimum = solver.Value; // should be 0

// Which can be double-checked against Wolfram Alpha if there is need:
// https://www.wolframalpha.com/input/?i=min+10+*+(x%2B1)%5E2+%2B+y%5E2

Reference

Accord.Math.Optimization Namespace