ConjGrad Routines

Summary

Minimizes the function of several variables by using the Conjugate gradient optimization algorithm.

Declaration

Function ConjGrad(Fun: TRealFunction; Grad: TGrad; var Pars: array of TSample; const Consts: array of TSample; const ObjConst: array of TObject; out FMin: TSample; out StopReason: TOptStopReason; FletcherAlgo: Boolean; SoftLineSearch: boolean): Integer;

Declaration

Parameter	Description
Fun	Real function (must be of TRealFunction type) to be minimized.
Grad	The gradient and Hessian procedure (must be of TGrad type), used for calculating the gradient.
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).
FletcherAlgo	If True, ConjGrad procedure will use Fletcher-Reeves method. If false, ConjGrad procedure will use Polal-Ribiere method.
SoftLineSearch	If True, ConjGrad internal line search algoritm will use soft line search method. Set SoftLineSearch to true if you're using numerical approximation for gradient. If SoftLineSearch if false, ConjGrad internal line search algorithm will use exact line search method. Set SoftLineSearch to false if you're using exact gradient.
MaxIter	Maximum allowed numer of minimum search iterations.
Tol	Desired Pars - minimum position tolerance.
GradTol	Minimum allowed gradient C-Norm.
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 Conjugate gradient optimization algorithm.

See Also
TGrad
TRealFunction
NumericGradDifference
NumericGradRichardson

Example 1

Problem: Find the minimum of the "Banana" function by using the Conjugate gradient method.
Solution:The Banana function is defined by the following equation:

Normally ConjGrad method would also require gradient procedure. But in this example we'll use the numerical approximation, more precisely the NumeridGradRichardson routine. This is done by specifying NumericGradRichardson routine as Grad parameter in ConjGrad routine call (see bellow)

    Uses MtxVec, Math387, Optimization;

    function Banana(Pars: TVec; Const Consts : Array of TSample;
    Const OConsts: Array of TObject): TSample;
    begin
      Result := 100*Sqr(Pars[1]-Sqr(Pars[0]))+Sqr(1-Pars[0]);
    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 := ConjGrad(Banana,NumericGradRichardson,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"
    #include "MtxIntDiff.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);
    }

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

    // Objective function
    private double Banana(TVec x, double[] c, object[] o)
    {
       return 100*Math387.IntPower(x[1]-Math387.IntPower(x[0],2),2) + Math387.IntPower(1-x[0],2);
    }

    private void Example()
    {
      double[2] Pars;
      double fmin;
      Matrix iHess;
      TOptStopReason StopReason;
      // initial estimates for x1 and x2
      Pars[0] = 0;
      Pars[1] = 0;
      int iters = BFGS(Banana,MtxIntDiff.NumericGradRichardson,Pars,null,null,out fmin, out StopReason,
                      true,false,1000,1.0e-8,1.0e-8,null);
      // stop if Iters >1000 or Tolerance < 1e-8
    }

Declaration