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{ ******************************************************************
Function minimization by the simplex method
****************************************************************** }
unit usimplex;
interface
uses
utypes;
procedure SaveSimplex(FileName : string);
{ ------------------------------------------------------------------
Opens a file to save the Simplex iterations
------------------------------------------------------------------ }
procedure Simplex(Func : TFuncNVar;
X : PVector;
Lb, Ub : Integer;
MaxIter : Integer;
Tol : Float;
var F_min : Float);
{ ------------------------------------------------------------------
Minimization of a function of several variables by the
simplex method of Nelder and Mead
------------------------------------------------------------------
Input parameters : Func = objective function
X = initial minimum coordinates
Lbound,
Ubound = indices of first and last variables
MaxIter = maximum number of iterations
Tol = required precision
------------------------------------------------------------------
Output parameters : X = refined minimum coordinates
F_min = function value at minimum
------------------------------------------------------------------
The function MathErr returns one of the following codes:
OptOk = no error
OptNonConv = non-convergence
------------------------------------------------------------------ }
implementation
const
WriteLogFile : Boolean = False;
var
LogFile : Text;
procedure SaveSimplex(FileName : string);
begin
Assign(LogFile, FileName);
Rewrite(LogFile);
WriteLogFile := True;
end;
procedure Simplex(Func : TFuncNVar;
X : PVector;
Lb, Ub : Integer;
MaxIter : Integer;
Tol : Float;
var F_min : Float);
const
Step = 1.50; { Step used to construct the initial simplex }
var
P : PMatrix; { Simplex coordinates }
F : PVector; { Function values }
Pbar : PVector; { Centroid coordinates }
Pstar, P2star : PVector; { New vertices }
Ystar, Y2star : Float; { New function values }
F0 : Float; { Function value at minimum }
N : Integer; { Number of parameters }
M : Integer; { Index of last vertex }
L, H : Integer; { Vertices with lowest & highest F values }
I, J : Integer; { Loop variables }
Iter : Integer; { Iteration count }
Corr, MaxCorr : Float; { Corrections }
Sum : Float;
Flag : Boolean;
procedure UpdateSimplex(Y : Float; Q : PVector);
{ Update "worst" vertex and function value }
var
J : Integer;
begin
F^[H] := Y;
for J := Lb to Ub do
P^[H]^[J] := Q^[J];
end;
begin
{ Quit if no iteration required }
if MaxIter < 1 then
begin
F_min := Func(X);
SetErrCode(OptOk);
Exit;
end;
if WriteLogFile then
begin
WriteLn(LogFile, 'Simplex');
WriteLn(LogFile, 'Iter F');
end;
N := Ub - Lb + 1;
M := Ub + 1;
DimMatrix(P, M, Ub);
DimVector(F, M);
DimVector(Pbar, Ub);
DimVector(Pstar, Ub);
DimVector(P2star, Ub);
Iter := 1;
F0 := MaxNum;
{ Construct initial simplex }
for I := Lb to M do
for J := Lb to Ub do
P^[I]^[J] := X^[J];
for I := Lb to Ub do
P^[I]^[I] := P^[I]^[I] * Step;
{ Evaluate function at each vertex }
for I := Lb to M do
F^[I] := Func(P^[I]);
repeat
{ Find vertices (L,H) having the lowest and highest
function values, i.e. "best" and "worst" vertices }
L := Lb;
H := Lb;
for I := Lb + 1 to M do
if F^[I] < F^[L] then
L := I
else if F^[I] > F^[H] then
H := I;
if F^[L] < F0 then
F0 := F^[L];
if WriteLogFile then
WriteLn(LogFile, Iter:4, ' ', F0:12);
{ Find centroid of points other than P(H) }
for J := Lb to Ub do
begin
Sum := 0.0;
for I := Lb to M do
if I <> H then Sum := Sum + P^[I]^[J];
Pbar^[J] := Sum / N;
end;
{ Reflect worst vertex through centroid }
for J := Lb to Ub do
Pstar^[J] := 2.0 * Pbar^[J] - P^[H]^[J];
Ystar := Func(Pstar);
{ If reflection successful, try extension }
if Ystar < F^[L] then
begin
for J := Lb to Ub do
P2star^[J] := 3.0 * Pstar^[J] - 2.0 * Pbar^[J];
Y2star := Func(P2star);
{ Retain extension or contraction }
if Y2star < F^[L] then
UpdateSimplex(Y2star, P2star)
else
UpdateSimplex(Ystar, Pstar);
end
else
begin
I := Lb;
Flag := False;
repeat
if (I <> H) and (F^[I] > Ystar) then Flag := True;
Inc(I);
until Flag or (I > M);
if Flag then
UpdateSimplex(Ystar, Pstar)
else
begin
{ Contraction on the reflection side of the centroid }
if Ystar <= F^[H] then
UpdateSimplex(Ystar, Pstar);
{ Contraction on the opposite side of the centroid }
for J := Lb to Ub do
P2star^[J] := 0.5 * (P^[H]^[J] + Pbar^[J]);
Y2star := Func(P2star);
if Y2star <= F^[H] then
UpdateSimplex(Y2star, P2star)
else
{ Contract whole simplex }
for I := Lb to M do
for J := Lb to Ub do
P^[I]^[J] := 0.5 * (P^[I]^[J] + P^[L]^[J]);
end;
end;
{ Test convergence }
MaxCorr := 0.0;
for J := Lb to Ub do
begin
Corr := Abs(P^[H]^[J] - P^[L]^[J]);
if Corr > MaxCorr then MaxCorr := Corr;
end;
Inc(Iter);
until (MaxCorr < Tol) or (Iter > MaxIter);
for J := Lb to Ub do
X^[J] := P^[L]^[J];
F_min := F^[L];
DelMatrix(P, M, Ub);
DelVector(F, M);
DelVector(Pbar, Ub);
DelVector(Pstar, Ub);
DelVector(P2star, Ub);
if WriteLogFile then
Close(LogFile);
if Iter > MaxIter then
SetErrCode(OptNonConv)
else
SetErrCode(OptOk);
end;
end.
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