Page 344 - Applied Numerical Methods Using MATLAB

UNCONSTRAINED OPTIMIZATION 333

with
T T
g
[g n+1 − g n ] g n+1 g n+1 n+1
β n = (FR) or (PR) (7.1.19)
T T
g g n g g n
n n
Step 3. Update the approximate solution point to x k+1 = x(N), which is the
last temporary one.
Step 4.If x k ≈ x k−1 and f(x k ) ≈ f(x k−1 ), then declare x k to be the minimum
and terminate the procedure. Otherwise, increment k by one and go back
to Step 1.

function [xo,fo] = opt_conjg(f,x0,TolX,TolFun,alpha0,MaxIter,KC)
%KC = 1: Polak–Ribiere Conjugate Gradient method
%KC = 2: Fletcher–Reeves Conjugate Gradient method
if nargin < 7, KC = 0; end
if nargin < 6, MaxIter = 100; end
if nargin < 5, alpha0 = 10; end
if nargin < 4, TolFun = 1e-8; end
if nargin < 3, TolX = 1e-6; end
N = length(x0); nmax1 = 20; warning = 0; h = 1e-4; %dimension of variable
x = x0; fx = feval(f,x0); fx0 = fx;
fork=1: MaxIter
xk0 = x; fk0 = fx; alpha = alpha0;
g = grad(f,x,h); s = -g;
for n = 1:N
alpha = alpha0;
fx1 = feval(f,x + alpha*2*s); %trial move in search direction
for n1 = 1:nmax1 %To find the optimum step size by line search
fx2 = fx1; fx1 = feval(f,x+alpha*s);
if fx0 > fx1 + TolFun & fx1 < fx2 - TolFun %fx0 > fx1 < fx2
den = 4*fx1 - 2*fx0 - 2*fx2; num = den-fx0 + fx2; %Eq.(7.1.5)
alpha = alpha*num/den;
x = x+alpha*s; fx = feval(f,x);
break;
elseif n1 == nmax1/2
alpha = -alpha0; fx1 = feval(f,x + alpha*2*s);
else
alpha = alpha/2;
end
end
x0 = x; fx0 = fx;
ifn<N
g1 = grad(f,x,h);
if KC <= 1, s=-g1 +(g1 - g)*g1’/(g*g’+ 1e-5)*s; %(7.1.19a)
else s = -g1 + g1*g1’/(g*g’+ 1e-5)*s; %(7.1.19b)
end
g = g1;
end
if n1 >= nmax1, warning = warning+1; %can’t find optimum step size
else warning = 0;
end
end
if warning >= 2|(norm(x - xk0)<TolX&abs(fx - fk0)< TolFun), break; end
end
xo=x; fo=fx;
if k == MaxIter, fprintf(’Just best in %d iterations’,MaxIter), end
%nm716 to minimize f(x) by the conjugate gradient method.
f713 = inline(’x(1).^2 - 4*x(1) - x(1).*x(2) + x(2).^2 - x(2)’,’x’);
x0 =[0 0], TolX = 1e-4; TolFun = 1e-4; alpha0 = 10; MaxIter = 100;
[xo,fo] = opt_conjg(f713,x0,TolX,TolFun,alpha0,MaxIter,1)
[xo,fo] = opt_conjg(f713,x0,TolX,TolFun,alpha0,MaxIter,2)