Optimization.BFGS Function

Minimizes the function of several variables by using the Quasi-Newton optimization algorithm.

Pascal

function BFGS(Fun: TRealFunction; Grad: TGrad; var Pars: Array of double; Const Consts: array of double; Const ObjConst: Array of TObject; out FMin: double; const IHess: TMtx; out StopReason: TOptStopReason; const FloatPrecision: TMtxFloatPrecision; DFPAlgo: Boolean = false; SoftLineSearch: boolean = true; MaxIter: Integer = 500; Tol: double = 1.0E-8; GradTol: double = 1.0E-8; const Verbose: TStrings = nil): Integer; overload;

File

Optimization

Parameters

Parameters	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	Additional Fun constant parameteres (can be/is usually nil).
ObjConst	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).
FloatPrecision	Specifies the floating point precision to be used by the routine.
DFPAlgo	If True, BFGS procedure will use Davidon-Fletcher-Powell Hessian update scheme. If False, BFGS procedure will use Broyden-Fletcher-Goldberg-Shamo Hessian update scheme.
SoftLineSearch	If True, BFGS 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, BFGS 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. 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 BFGS-DFP method.

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

Also, BFGS method requires the gradient of the function. The gradient of the Banana function is:

Uses MtxVec, Math387, Optimization, MtxIntDiff; function Banana(const Pars: TVec; const Consts: TVec; const OConsts: Array of TObject): double; begin Banana := 100*Sqr(Pars[1]-Sqr(Pars[0]))+Sqr(1-Pars[0]); end; procedure GradBanana(Fun: TRealFunction; const Pars: TVec; const Consts: TVec; const ObjConsts: Array of TObject; const Grad: TVec); begin Grad[0] := -400*(Pars[1] - Sqr(Pars[0]))*Pars[0] - 2*(1 - Pars[0]); Grad[1] := 200*(Pars[1] - Sqr(Pars[0])); end; procedure Example; var Iters : integer; Pars : Array [0..1] of double; StopReason : TOptStopReason; begin // initial estimates for x1 and x2 Pars[0] := 0; Pars[1] := 0; Iters := BFGS(Banana,GradBanana,Pars,[],[],FMin,StopReason,mvDouble,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 "MtxExpr.hpp" #include "Math387.hpp" #include "Optimization.hpp" #include "MtxIntDiff.hpp" // Objective function double __fastcall Banana(TVec* const Parameters, TVec* const Constants, 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 of the objective function void __fastcall GradBanana(TRealFunction Fun, TVec* const Parameters, TVec* const Consts, System::TObject* const * ObjConst, const int PConsts_Size, Mtxvec::TVec* const Grad) { 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)); } void __fastcall Example(); { double Pars[2]; double fmin; TOptStopReason StopReason; // initial estimates for x1 and x2 Pars[0] = 0; Pars[1] = 0; int iters = BFGS(Banana,GradBanana,Pars,1,NULL,-1,NULL,-1,fmin,iHess, StopReason,mvDouble, false,true,1000,1.0e-8,1.0e-8,NULL); // stop if Iters >1000 or Tolerance < 1e-8 }

Optimization, Functions, Example, See Also

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