Page 359 - Applied Numerical Methods Using MATLAB

348 OPTIMIZATION
   
0
−x 1
0
−x 2
   
   
s.t. g(x) =  3x 1 − x 1 x 2 + 4x 2 − 7  ≤  0  (E7.3.1b)


 
   
2x 1 + x 2 − 3
   0 
2
3x 1 − 4x − 4x 2 0
2
According to the penalty function method, we construct a new objective func-
tion (7.2.6) as
2 2 2 2
Min l(x) ={(x 1 + 1.5) + 5(x 2 − 1.7) }{(x 1 − 1.4) + 0.6(x 2 − 0.5) }
5

+ v m ψ m (g m (x)) (E7.3.2a)
m=1
where
0 if g m (x) ≤ 0 (constraint satisfied)

v m = 1, ψ m (g m (x)) = ,
exp(e m g m (x)) if g m (x)> 0 (constraint violated)
e m = 1 ∀ m = 1,..., 5 (E7.3.2b)

%nm722 for Ex.7.3
% to solve a constrained optimization problem by penalty ftn method.
clear, clf
f =’f722p’;
x0=[0.4 0.5]
TolX = 1e-4; TolFun = 1e-9; alpha0 = 1;
[xo_Nelder,fo_Nelder] = opt_Nelder(f,x0) %Nelder method
[fc_Nelder,fo_Nelder,co_Nelder] = f722p(xo_Nelder) %its results
[xo_s,fo_s] = fminsearch(f,x0) %MATLAB built-in fminsearch()
[fc_s,fo_s,co_s] = f722p(xo_s) %its results
% including how the constraints are satisfied or violated
xo_steep = opt_steep(f,x0,TolX,TolFun,alpha0) %steepest descent method
[fc_steep,fo_steep,co_steep] = f722p(xo_steep) %its results
[xo_u,fo_u] = fminunc(f,x0); % MATLAB built-in fminunc()
[fc_u,fo_u,co_u] = f722p(xo_u) %its results
function [fc,f,c] = f722p(x)
f=((x(1)+ 1.5)^2 + 5*(x(2)- 1.7)^2)*((x(1)- 1.4)^2 + .6*(x(2)-.5)^2);
c=[-x(1); -x(2); 3*x(1) - x(1)*x(2) + 4*x(2) - 7;
2*x(1)+ x(2) - 3; 3*x(1) - 4*x(2)^2 - 4*x(2)]; %constraint vector
v=[11111]; e=[1111 1]’; %weighting coefficient vector
fc = f +v*((c > 0).*exp(e.*c)); %new objective function