Marquardt Routines

Summary

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

Declaration

Function Marquardt(Fun: TRealFunction; GradHess: TGradHess; var Pars: array of TSample; const Consts: array of TSample; const ObjConst: array of TObject; out FMin: TSample; IHess: TMtx; out StopReason: TOptStopReason; MaxIter: Integer = 500; Tol: TSample = 1.0E-8; GradTol: TSample = 1.0E-8; Lambda0: TSample = 1e-2; Verbose: TStrings = nil): Integer;

Parameter	Description
Fun	Real function (must be of TRealFunction type) to be minimized.
GradHess	The gradient and Hessian procedure (must be of TGradHess type), used for calculating the gradient and Hessian matrix.
Pars	Stores the initial estimates for parameters (minimum estimate). After the call to routine returns adjusted calculated values (minimum position).
Consts,PConsts	Additional Fun constant parameteres (can be/is usually nil).
FMin	Returns function value at minimum.
IHess	Returns inverse Hessian matrix.
StopReason	Returns reason why minimum search stopped (see TOptStopReason).
MaxIter	Maximum allowed numer of minimum search iterations.
Tol	Desired Pars - minimum position tolerance.
GradTol	Minimum allowed gradient C-Norm.
Lambda0	Initial lambda step, used in Marquardt algorithm.
Verbose	If assigned, stores Fun, evaluated at each iteration step.

Result

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

Description

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

See Also
TGradHess
TRealFunction
NumericGradHess

Example 1

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 :

    Uses MtxVec, Math387, Optimization;

    function Banana(Pars: TVec; const Consts : Array of TSample;  const PConsts: Array of TObject): TSample;
    begin
      Result := 100*Sqr(Pars[1]-Sqr(Pars[0]))+Sqr(1-Pars[0]);
    end;
    procedure GradHessBanana(Fun: TRealFunction; Pars: TVec; const consts: Array of TSample;
    Const PConsts: Array of TObject; Grad: TVec; Hess: TMtx);
    begin
      Grad.Values[0] := -400*(Pars[1]-Sqr(Pars[0]))*Pars[0]-2*(1-Pars[0]);
      Grad.Values[1] := 200*(Pars[1]-Sqr(Pars[0]));
      Hess.Values[0,0] := -400*Pars[1]+1200*Sqr(Pars[0])+2;
      Hess.Values[0,1] := -400*Pars[0];
      Hess.Values[1,0] := Hess.Values[0,1]; // symmetric !
      Hess.Values[1,1] := 200;
    end;

    procedure Example;
    var Iters : integer;
      Pars : Array [0..1] of TSample;
      StopReason : TOptStopReason;
    begin
      // initial estimates for x1 and x2
      Pars[0] := 0;
      Pars[1] := 0;
      Iters := Marquardt(Banana,GradHessBanana,Pars,[],[],FMin,StopReason,IHess);
      //stop if Iters > 500 or Tolerance < 1e-8
      // Returns Pars = [1,1] and FMin = 0, meaning x1=1, x2=1 and minimum value is 0
    end;

    #include "MtxVecCpp.h" //MtxVecCPP.cpp must be included in the project
    #include "Math387.hpp"
    #include "Optimization.hpp"

    // Objective function
    double __fastcall Banana(TVec * Parameters, const double * Constants, const int Constants_Size,
     System::TObject* const * ObjConst, const int ObjConst_Size)
    {
      double* Pars = Parameters->PValues1D(0);

      return 100.0*IntPower(Pars[1]-IntPower(Pars[0],2),2)+IntPower(1.0-Pars[0],2);
    }
    // Analytical gradient and Hessian matrix of the objective function
    void __fastcall BananaGradHess(TRealFunction Fun, TVec * Paraneters, const double * Consts, const int Consts_Size,
	    System::TObject* const * ObjConst, const int ObjConst_Size, TVec* Grad, TMtx* Hess)
    {
      double* Pars = Parameters->PValues1D(0);

      Grad->Values[0] = -400*(Pars[1]-IntPower(Pars[0],2))*Pars[0] - 2*(1-Pars[0]);
      Grad->Values[1] = 200*(Pars[1]-IntPower(Pars[0],2));
      Hess->Values1D[0] = -400*Pars[1]+1200*IntPower(Pars[0],2)+2;
      Hess->Values1D[1] = -400*Pars[0];
      Hess->Values1D[2] = -400*Pars[0];
      Hess->Values1D[3] = 200;
    }

    void __fastcall Example;
    {
      double Pars[2];
      double fmin;
      Matrix iHess;
      TOptStopReason StopReason;
      // initial estimates for x1 and x2
      Pars[0] = 0;
      Pars[1] = 0;
      int iters = Marquardt(Banana,BananaGradHess,Pars,1,NULL,-1,NULL,-1,fmin, iHess,
                      StopReason, 1000,1.0e-8,1.0e-3,NULL);
      // stop if Iters >1000 or Tolerance < 1e-8
    }

Declaration

Function Marquardt(Fun: TRealFunction; GradHess: TGradHess; var Pars: array of TSample; const Consts: array of TSample; const ObjConst: array of TObject; out FMin: TSample; IHess: TMtx; out StopReason: TOptStopReason; MaxIter: Integer; Tol: TSample; GradTol: TSample; Lambda0: TSample): Integer;

Declaration