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Showing content from http://accord-framework.net/docs/html/T_Accord_Math_Optimization_Cobyla.htm below:

Constrained optimization by linear approximation.

Accord.Math (in Accord.Math.dll) Version: 3.8.0

public class Cobyla : BaseOptimizationMethod, IOptimizationMethod, 
	IOptimizationMethod<double[], double>, IOptimizationMethod<CobylaStatus>, 
	IOptimizationMethod<double[], double, CobylaStatus>

Public Class Cobyla
	Inherits BaseOptimizationMethod
	Implements IOptimizationMethod, IOptimizationMethod(Of Double(), Double), 
	IOptimizationMethod(Of CobylaStatus), IOptimizationMethod(Of Double(), Double, CobylaStatus)

The Cobyla type exposes the following members.

Determines whether the specified object is equal to the current object.

Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection.

Serves as the default hash function.

Gets the Type of the current instance.

Finds the maximum value of a function. The solution vector will be made available at the

property.

Finds the maximum value of a function. The solution vector will be made available at the

property.

Creates a shallow copy of the current Object.

Finds the minimum value of a function. The solution vector will be made available at the

property.

Finds the minimum value of a function. The solution vector will be made available at the

property.

Called when the

property has changed.

Implements the actual optimization algorithm. This method should try to minimize the objective function.

Returns a string that represents the current object.

Checks whether an object implements a method with the given name.

Compares two objects for equality, performing an elementwise comparison if the elements are vectors or matrices.

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.

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.

Constrained optimization by linear approximation (COBYLA) is a numerical optimization method for constrained problems where the derivative of the objective function is not known, invented by Michael J. D. Powell.

COBYLA2 is an implementation of Powell’s nonlinear derivative–free constrained optimization that uses a linear approximation approach. The algorithm is a sequential trust–region algorithm that employs linear approximations to the objective and constraint functions, where the approximations are formed by linear interpolation at n + 1 points in the space of the variables and tries to maintain a regular–shaped simplex over iterations.

This algorithm is able to solve non-smooth NLP problems with a moderate number of variables (about 100), with inequality constraints only.

References:

• Wikipedia, The Free Encyclopedia. Cobyla. Available on: http://en.wikipedia.org/wiki/COBYLA

Let's say we would like to optimize a function whose gradient we do not know or would is too difficult to compute. All we have to do is to specify the function, pass it to Cobyla and call its Minimize() method:

Func<double[], double> function = x => 10 * Math.Pow(x[0] + 1, 2) + Math.Pow(x[1], 2);


Cobyla cobyla = new Cobyla(2, function);

bool success = cobyla.Minimize();

double minimum = minimum = cobyla.Value; 
double[] solution = cobyla.Solution;     

Cobyla can be used even when we have constraints in our optimization problem. The following example can be found in Fletcher's book Practical Methods of Optimization, under the equation number (9.1.15).

var f = new NonlinearObjectiveFunction(2, x => -x[0] - x[1]);


var constraints = new[]
{
    new NonlinearConstraint(2, x =>             x[1] - x[0] * x[0] >= 0),
    new NonlinearConstraint(2, x =>  1 - x[0] * x[0] - x[1] * x[1] >= 0),
};


var cobyla = new Cobyla(function, constraints);


bool success = cobyla.Minimize();
double minimum = cobyla.Value;        
double[] solution = cobyla.Solution;  

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