Page 362 - Applied Numerical Methods Using MATLAB

MATLAB BUILT-IN ROUTINES FOR OPTIMIZATION 351

%nm731_1
% to minimize an objective function f(x) by various methods.
clear, clf
% An objective function and its gradient function
f = inline(’(x(1) - 0.5).^2.*(x(1) + 1).^2 + (x(2)+1).^2.*(x(2) - 1).^2’,’x’);
g0 = ’[2*(x(1)- 0.5)*(x(1)+ 1)*(2*x(1)+ 0.5) 4*(x(2)^2 - 1).*x(2)]’;
g = inline(g0,’x’);
x0 = [0 0.5] %initial guess
[xon,fon] = opt_Nelder(f,x0) %min point, its ftn value by opt_Nelder
[xos,fos] = fminsearch(f,x0) %min point, its ftn value by fminsearch()
[xost,fost] = opt_steep(f,x0) %min point, its ftn value by opt_steep()
TolX = 1e-4; MaxIter = 100;
xont = Newtons(g,x0,TolX,MaxIter);
xont,f(xont) %minimum point and its function value by Newtons()
[xocg,focg] = opt_conjg(f,x0) %min point, its ftn value by opt_conjg()
[xou,fou] = fminunc(f,x0) %min point, its ftn value by fminunc()

Noting that it depends mainly on the initial value x 0 whether each routine
succeeds in finding a minimum point, we summarize the results of running those
routines with various initial values in Table 7.2. It can be seen from this table
that the gradient-based optimization routines like “opt_steep()”, “Newtons()”,
“opt_conj()”, and “fminunc()” sometimes get to a saddle point or even a
maximum point (Remark 7.1) and that the routines do not always approach the
extremum that is closest to the initial point. It is interesting to note that even
the non-gradient-based MATLAB built-in routine “fminsearch()” may get lost,
while our routine “opt_Nelder()” works well for this case. We cannot, how-
ever, conclude that this routine is better than that one based on only one trial,
because there may be some problems for which the MATLAB built-in routine
works well, but our routine does not. What we can state over this happening is
that no human work is free from defect.
Now, we will see a MATLAB built-in routine “lsqnonlin(f,x0,l,u,
options,p1,..)”, which presents a nonlinear least-squares (NLLS) solution to

Table 7.2 Results of Running Several Unconstrained Optimization Routines with
Various Initial Values

x 0 opt − Nelder fminsearch opt − steep Newtons opt − conjg fminunc
[0, 0] [−1, 1] [0.5, 1] [0.5, 0] [−0.25, 0] [0.5, 0] [0.5, 0]
(minimum) (minimum) (saddle) (maximum) (saddle) (saddle)
[0, 0.5] [0.5, 1] [0.02, 1] [0.5, 1] [−0.25, −1] [0.5, 1] [0.5, 1]
(minimum) (lost) (minimum) (saddle) (minimum) (minimum)
[0.4, 0.5] [0.5, 1] [0.5, 1] [0.5, 1] [0.5, −1] [0.5, 1] [0.5, 1]
(minimum) (minimum) (minimum) (minimum) (minimum) (minimum)
[−0.5, 0.5] [0.5, 1] [−1, 1] [−1, 1] [−0.25, −1] [−1, 1] [−1, 1]
(minimum) (minimum) (minimum) (saddle) (minimum) (minimum)
[−0.8, 0.5] [−1, 1] [−1, 1] [−1, 1] [−1, −1] [−1, 1] [−1, 1]
(minimum) (minimum) (minimum) (minimum) (minimum) (minimum)