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QuickStart Samples

# Optimization In Multiple Dimensions QuickStart Sample (C#)

Illustrates the use of the multi-dimensional optimizer classes in the Extreme.Mathematics.Optimization namespace for optimization in multiple dimensions in C#.

```using System;

namespace Extreme.Numerics.QuickStart.CSharp
{
// The optimization classes reside in the
// Extreme.Mathematics.Optimization namespace.
using Extreme.Mathematics.Optimization;
// Function delegates reside in the Extreme.Mathematics
// namespace.
using Extreme.Mathematics;
// Vectors reside in the Extreme.Mathematics.LinearAlgebra
// namespace.
using Extreme.Mathematics.LinearAlgebra;

/// <summary>
/// Illustrates unconstrained optimization in multiple dimensions
/// using classes in the Extreme.Mathematics.Optimization
/// namespace of the Extreme Optimization Numerical Libraries
/// for .NET.
/// </summary>
class OptimizationInND
{
/// <summary>
/// The main entry point for the application.
/// </summary>
static void Main(string[] args)
{
//
// Objective function
//

// The objective function must be supplied as a
// Func<Vector<double>, double> delegate. This is a method
// that takes one var argument and returns a real number.
// See the end of this sample for definitions of the methods
// that are referenced here.
Func<Vector<double>, double> f = fRosenbrock;

// The gradient of the objective function can be supplied either
// as a MultivariateVectorFunction delegate, or a
// MultivariateVectorFunction delegate. The former takes
// one vector argument and returns a vector. The latter
// takes a second vector argument, which is an existing
// vector that is used to return the result.
Func<Vector<double>, Vector<double>, Vector<double>> g = gRosenbrock;

// The initial values are supplied as a vector:
var initialGuess = Vector.Create(-1.2, 1);
// The actual solution is [1, 1].

//
// Quasi-Newton methods: BFGS and DFP
//

// For most purposes, the quasi-Newton methods give
// excellent results. There are two variations: DFP and
// BFGS. The latter gives slightly better results.

// Which method is used, is specified by a constructor
// parameter of type QuasiNewtonMethod:
var bfgs = new QuasiNewtonOptimizer(QuasiNewtonMethod.Bfgs);

bfgs.InitialGuess = initialGuess;
bfgs.ExtremumType = ExtremumType.Minimum;

// Set the ObjectiveFunction:
bfgs.ObjectiveFunction = f;
// The FindExtremum method does all the hard work:
bfgs.FindExtremum();

Console.WriteLine("BFGS Method:");
Console.WriteLine("  Solution: {0}", bfgs.Extremum);
Console.WriteLine("  Estimated error: {0}", bfgs.EstimatedError);
Console.WriteLine("  # iterations: {0}", bfgs.IterationsNeeded);
// Optimizers return the number of function evaluations
// and the number of gradient evaluations needed:
Console.WriteLine("  # function evaluations: {0}", bfgs.EvaluationsNeeded);

//
//

// Conjugate gradient methods exist in three variants:
// Fletcher-Reeves, Polak-Ribiere, and positive Polak-Ribiere.

// Which method is used, is specified by a constructor
// Everything else works as before:
cg.ObjectiveFunction = f;
cg.InitialGuess = initialGuess;
cg.FindExtremum();

Console.WriteLine("  Solution: {0}", cg.Extremum);
Console.WriteLine("  Estimated error: {0}", cg.EstimatedError);
Console.WriteLine("  # iterations: {0}", cg.IterationsNeeded);
Console.WriteLine("  # function evaluations: {0}", cg.EvaluationsNeeded);

//
// Powell's method
//

// Powell's method is a conjugate gradient method that
// does not require the derivative of the objective function.
// It is implemented by the PowellOptimizer class:
var pw = new PowellOptimizer();
pw.InitialGuess = initialGuess;
// Powell's method does not use derivatives:
pw.ObjectiveFunction = f;
pw.FindExtremum();

Console.WriteLine("Powell's Method:");
Console.WriteLine("  Solution: {0}", pw.Extremum);
Console.WriteLine("  Estimated error: {0}", pw.EstimatedError);
Console.WriteLine("  # iterations: {0}", pw.IterationsNeeded);
Console.WriteLine("  # function evaluations: {0}", pw.EvaluationsNeeded);

//
//

// Also called the downhill simplex method, the method of Nelder
// and Mead is useful for functions that are not tractable
// by other methods. For example, other methods
// may fail if the objective function is not continuous.
// Otherwise it is much slower than other methods.

// The method is implemented by the NelderMeadOptimizer class:

// The class has three special properties, that help determine
// the progress of the algorithm. These parameters have
// default values and need not be set explicitly.
nm.ContractionFactor = 0.5;
nm.ExpansionFactor = 2;
nm.ReflectionFactor = -2;

// Everything else is the same.
nm.SolutionTest.AbsoluteTolerance = 1e-15;
nm.InitialGuess = initialGuess;
// The method does not use derivatives:
nm.ObjectiveFunction = f;
nm.FindExtremum();

Console.WriteLine("  Solution: {0}", nm.Extremum);
Console.WriteLine("  Estimated error: {0}", nm.EstimatedError);
Console.WriteLine("  # iterations: {0}", nm.IterationsNeeded);
Console.WriteLine("  # function evaluations: {0}", nm.EvaluationsNeeded);

Console.Write("Press Enter key to exit...");
}

// The famous Rosenbrock test function.
static double fRosenbrock(Vector<double> x)
{
double p = (1-x[0]);
double q = x[1] - x[0]*x[0];
return p*p + 105 * q*q;
}

// Gradient of the Rosenbrock test function.
static Vector<double> gRosenbrock(Vector<double> x, Vector<double> f)
{
// Always assume that the second argument may be null:
if (f == null)
f = Vector.Create<double>(2);
double p = (1-x[0]);
double q = x[1] - x[0]*x[0];
f[0] = -2*p - 420*x[0]*q;
f[1] = 210*q;
return f;
}
}
}```