Optimization.TrustRegion Function

Trust region algorithm for finding minimum of vector function.

Pascal

function TrustRegion(Fun: TVectorFunction; JacProc: TJacobianFunction; const X: TVec; const Y: TVec; Const Consts: Array of double; Const OConsts: Array of TObject; MaxIter: Integer; MaxTrialIter: Integer; Const EPSArray: TEPSArray; Rs: double; out StopReason: TOptStopReason; const Control: TOptControl = nil): Integer; overload;

File

Optimization

Parameters

Parameters	Description
Fun	The objective function to be minimized.
JacProc	The Jacobian matrix calculation procedure. If it is nil, then the numerical approximation will be used to evaluate Jacobi matrix elements.
X	Stores the initial estimates for X. On completion returns estimates, evaluated at function minimum.
Y	Returns objective function value, evaluated at minimum.
Consts	Additional Fun constant parameteres (can be/is usually nil).
OConsts	Additional Fun constant parameteres (can be/is usually nil).
MaxIter	Defines the maximum number of main TR algorithm loops.
MaxTrialIter	Defines the maximum number of iterations of trial-step calculation.
EPSArray	Array of size 6. Contains stopping tests. eps[0]: Grad < eps[0] eps[1]: \|\|F(x)\|\| < eps[1] eps[2]: \|\|A(x)ij\|\| < eps[2] eps[3]: \|\|s\|\| < eps[3] eps[4]: \|\|F(x)\|\|- \|\|F(x) - A(x)s\|\| < eps[4] eps[5]: trial step precision. If eps[5] = 0, then eps[5] = 1E-10, where A(x) = the jacobi matrix F(x) = \|\|y - f(x)\|\|
Rs	Positive input variable used in determining the initial step bound. In most cases the factor should lie within the interval (0.1, 100.0). The generally recommended value is 100.
StopReason	Returns the TR algorithm stop reason.
Control	Optional object, which allows the interruption of the search.

Returns

the number of iterations needed for results.

Description

The Trust Region (TR) algorithms are relatively new iterative algorithms for solving nonlinear optimization problems. They are widely used in power engineering, finance, applied mathematics, physics, computer science, economics, sociology, biology, medicine, mechanical engineering, chemistry, and other areas. TR methods have global convergence and local super convergence, which differenciates them from line search methods and Newton methods. TR methods have better convergence when compared with widely-used Newton-type methods.

The main idea behind TR algorithm is calculating a trial step and checking if the next values of x belong to the trust region. Calculation of the trial step is strongly associated with the approximation model.

For more on TR algorithm, check the following links:

http://iridia.ulb.ac.be/~fvandenb/optimization/optimizationIntro.html
http://en.wikipedia.org/wiki/Trust_region

Notes

This function is currently not supported under .NET. The objective function calls are threaded. The objective function should not be allocating any memory on the heap because calling threads are created outside of Delphi/BCB code.

Example

Uses MtxExpr, Optimization, MtxVec, Math387; // Objective function, 4 variables, 4 f components procedure TestVFun(const x,f: TVec; const c: Array of double; const o: Array of TObject); begin f[0] := x[0] + 10.0*x[1]; f[1] := 2.2360679774997896964091736687313*(x[2] - x[3]); f[2] := (x[1] - 2.0*x[2])*(x[1] - 2.0*x[2]); f[3] := 3.1622776601683793319988935444327*(x[0] - x[3])*(x[0] - x[3]); end; procedure Example; var x,f: Vector; epsa: TEPSArray; sr: TOptStopReason; begin // Initial estimates for variabless x.SetIt(false,[3,-1,0,1]); // 4 components, size must match the TestVFun implementation above f.Size(4); // setup stopping criteria, use default values epsa[0] := 1.0E-5; epsa[1] := 1.0E-5; epsa[2] := 1.0E-5; epsa[3] := 1.0E-5; epsa[4] := 1.0E-5; epsa[5] := 1.0E-10; // Minimize TrustRegion(TestVFun,x,f,[],[],1000,100,epsa,0.0,sr); // x stores minimum position (variables) // f stores function value at minimum // sr stores stop reason end;

#include "MtxExpr.hpp" #include "Math387.hpp" #include "Optimization.hpp" void __fastcall TestVFun(TVec* const x, TVec* const f, const double * c, const int c_Size, System::TObject* const * o, const int o_Size) { double* X = x->PValues1D(0); f->Values[0] = X[0] + 10.0*X[1]; f->Values[1] = 2.2360679774997896964091736687313*(X[2] - X[3]); f->Values[2] = (X[1] - 2.0*X[2])*(X[1] - 2.0*X[2]); f->Values[3] = 3.1622776601683793319988935444327*(X[0] - X[3])*(X[0] - X[3]); } void __fastcall Example(); { Vector x,f; TEPSArray epsarr; TOptStopReason sr; // Initial estimates for variabless x->SetIt(false,OPENARRAY(double,(3,-1,0,1))); // 4 components, size must match the TestVFun implementation above f->Size(4,false); // setup stopping criteria, use default values epsarr[0] = 1.0E-5; epsarr[1] = 1.0E-5; epsarr[2] = 1.0E-5; epsarr[3] = 1.0E-5; epsarr[4] = 1.0E-5; epsarr[5] = 1.0E-10; // Minimize TrustRegion(TestVFun,NULL,x,f,NULL,-1,NULL,-1,1000,100,epsarr,0.0,sr); // x stores minimum position (variables) // f stores function value at minimum // sr stores stop reason }

Optimization, Functions, Example

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