Optimization.Marquardt Method (TRealFunction, TGradHess, double[], [In] double[], [In] object[], out double, TMtx, out TOptStopReason, [In] TMtxFloatPrecision, int, double, double, double, TStrings)

Minimizes the function of several variables by using the Marquardt optimization algorithm.

Syntax

Visual Basic

public static int Marquardt(TRealFunction Fun, TGradHess GradHess, ref double[] Pars, [In] double[] Consts, [In] object[] ObjConst, out double FMin, TMtx IHess, out TOptStopReason StopReason, [In] TMtxFloatPrecision FloatPrecision, int MaxIter, double Tol, double GradTol, double Lambda0, TStrings Verbose);

Parameters

Parameters	Description
TRealFunction Fun	Real function (must be of TRealFunction type) to be minimized.
TGradHess GradHess	The gradient and Hessian procedure (must be of TGradHess type), used for calculating the gradient and Hessian matrix.
ref double[] Pars	Stores the initial estimates for parameters (minimum estimate). After the call to routine returns adjusted calculated values (minimum position).
[In] double[] Consts	Additional Fun constant parameteres (can be/is usually nil).
[In] object[] ObjConst	Additional Fun constant parameteres (can be/is usually nil).
out double FMin	Returns function value at minimum.
TMtx IHess	Returns inverse Hessian matrix.
out TOptStopReason StopReason	Returns reason why minimum search stopped (see TOptStopReason).
[In] TMtxFloatPrecision FloatPrecision	Specifies the floating point precision to be used by the routine.
int MaxIter	Maximum allowed numer of minimum search iterations.
double Tol	Desired Pars - minimum position tolerance.
double GradTol	Minimum allowed gradient C-Norm.
double Lambda0	Initial lambda step, used in Marquardt algorithm.
TStrings Verbose	If assigned, stores Fun, evaluated at each iteration step. Optionally, you can also pass TOptControl object to the Verbose parameter. This allows the optimization procedure to be interrupted from another thread and optionally also allows logging and iteration count monitoring.

Returns

the number of iterations required to reach the solution(minimum) within given tolerance.

Example

Problem: Find the minimum of the "Banana" function by using the Marquardt method.

Solution:The Banana function is defined by the following equation:

Also, Marquardt method requires the gradient and Hessian matrix of the function. The gradient of the Banana function is:

and the Hessian matrix is :

// Objective function private double Banana(TVec x, TVec c, params object[] o) { return 100*Math387.IntPower(x[1] - Math387.IntPower(x[0],2),2) + Math387.IntPower(1-x[0],2); } // Analytical gradient and Hessian matrix of the objective function private void BananaGradHess(TRealFunction Fun, TVec Pars, TVec Consts, object[] obj, TVec Grad, TMtx Hess) { double[] Pars = Parameters.PValues1D(0); Grad[0] = -400.0*(Pars[1]-Math387.IntPower(Pars[0],2))*Pars[0] - 2*(1-Pars[0]); Grad[1] = 200.0*(Pars[1]-Math387.IntPower(Pars[0],2)); Hess.Values1D[0] = -400.0*Pars[1]+1200*Math387.IntPower(Pars[0],2)+2; Hess.Values1D[1] = -400.0*Pars[0]; Hess.Values1D[2] = -400.0*Pars[0]; Hess.Values1D[3] = 200.0; } private void Example() { double[2] Pars; double fmin; Matrix iHess = new Matrix(0,0); TOptStopReason StopReason; // initial estimates for x1 and x2 Pars[0] = 0; Pars[1] = 0; int iters = Optimization.Marquardt(Banana,BananaGradHess,Pars, null, null, out fmin, iHess, out StopReason, TMtxFloatPrecision.mvDouble, 1000, 1.0e-8, 1.0e-8, null); // stop if Iters >1000 or Tolerance < 1e-8 }

Group

Marquardt Method

Links

Optimization Class, Optimization Members, Dew.Math.Units Namespace, Marquardt Method, Example, See Also

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