rosenbrock.m

• function [f,Df]=rosen(x)
  % the 2D Rosenbrock function and its gradient
  [m,n]=size(x);
  if (m ~= 2 | n ~= 1)
  	error('Bad data sent to 2D Rosenbrock function')
  end
  z    = [x(1)^2 - x(2), x(1)-1]';
  f    = 100*z(1)^2 + z(2)^2;
  Df   = [400*x(1)*z(1) + 2*z(2), -200*z(1)]';
  

Surface Plot of the Rosenbrock function

• [X,Y]=meshgrid(-2:0.1:2);
  Z=100*(Y-X.^2).^2+(ones(size(X))-X).^2;
  surf(X,Y,Z)
  

Contour Plot of the Rosenbrock function

• [X,Y]=meshgrid(-2:0.1:2);
  Z=100*(Y-X.^2).^2+(ones(size(X))-X).^2;
  contour(X,Y,Z)
  

Minimize Rosenbrock by Steepest Descent

minRosenBySD.m

%In this script we apply steepest descent with the
%backtracking linesearch to minimize the 2-D
%Rosenbrock function starting at the point x=(-1.9,2).


%Termination parameters
eps     =  1.0e-4;
epsf    =  1.0e-6;
maxit   =  10000;
iter    =  0;

%Linesearch parameters for backtracking
gamma   =  0.5;
c       =  0.01;

%Initialization
xc      =  [-1.9;2];
fnc     =  'rosenbrock';
[fc,Df] =  feval(fnc,xc);
nDf     =  norm(Df);

%Are we already at a solution, or should we continue.
if nDf <= epsf,
   disp('Termination due to small gradient.')
else

%Initialize main loop.
ndiff   =  1;

data    =  [iter,nDf,ndiff,1,fc];

%The main loop.
while nDf > epsf & ndiff > eps & iter < maxit,

        d       =  -Df/nDf;
        DDfnc   =  Df'*d;
        [xn,fn,fcall]=backtrack(xc,d,fc,fnc,DDfnc,c,gamma,eps);
        ndiff   =  norm(xn-xc);
        xc      =  xn;
        [fc,Df] =  feval(fnc,xc);
        nDf     =  norm(Df);
        iter    =  iter + 1;
        data    =  [data;[iter,nDf,ndiff,fcall,fc]];

%Report reason for termination.
        if nDf <= epsf,
           disp('Termination due to small gradient.')
           break
        elseif ndiff <= eps
           disp('Termination due to small steps.')
           break
        elseif iter == maxit
           disp('Exceeded maximum number of iterations.')
           break
        end
end

%Report on how the algorithm performed.
[datalength,width]   =  size(data);
onenorm              =  ones(datalength,1);
Tfcall               =  onenorm'*data(:,4);
disp('Final point is '), xc
disp('Final function value is'), fc
disp('Final norm of the gradient is '), nDf
disp('Total number of function calls is '), Tfcall
disp('Total number of gradient calls is '), iter
subplot(2,2,1), plot(data(:,4))
xlabel('iteration'),ylabel('function calls')
subplot(2,2,2), semilogy(data(:,1),data(:,2))
xlabel('iteration'),ylabel('norm of the gradient')
subplot(2,2,3), semilogy(data(:,1),data(:,3))
xlabel('iteration'),ylabel('distance between iterates')
subplot(2,2,4), semilogy(data(:,1),data(:,5))
xlabel('iteration'),ylabel('function value')
end