Golden Search Optimization Technique

Hi all. I am trying to find the maximum value of the function using the Golden Search algorithm. I have double-checked through my calculator, and the maximum value is at x=1.0158527. However, that is now what I get on my program. I am wondering if someone can help where I am going wrong.Here is the code and the results i get

%%%Author: Jacob Joshua Shila
%%%Golden Search Algorithm
clear all
clc 
format short e
syms x

%%Input
fx = 0.4/sqrt(1+x^2)-sqrt(1+x^2)*(1-.4/(1+x^2))+x;
maxit = 50;
es = 10^-5;
R = (5^.5-1)/2;

%%Determine the Interval for the Initial Guess
x=[-10:10];
f = subs(fx,x);
plot(x,f);
xlow = 0.5;
xhigh = 1.5;

%%Perform Golden Search
xl = xlow;
xu = xhigh;
iter = 1;
d = R*(xu-xl);
x1 = xl+d;
x2 = xu-d;
f1 = subs(fx,x1);
f2 = subs(fx,x2);

if f1>f2
    xopt = x1;
    fx = f1;
else
    xopt = x2;
    fx = f2;
end

while(1)
    d = R*d;
    if f1>f2
        xl = x2;
        x2 = x1;
        x1 = xl+d;
        f2 = f1;
        f1 = subs(fx,x1);
    else
        xu = x1;
        x1 = x2;
        x2 = xu-d;
        f1 = f2;
        f2 = subs(fx,x2);
    end
    iter = iter+1;
    if f1>f2
        xopt = x1;
        fx = f1;
    else
        xopt = x2;
        fx = f2;
    end
    if xopt~=0
        ea = (1-R)*abs((xu-xl)/xopt)*100;
    end
    if ea<=es||iter>=maxit,break
    end
end

Gold = xopt

and here is the results which I get from the program:

Gold =

    8.8197e-001

Thank you and any help appreciated.

Answer 1

[Alexander]

I'm not familiar with the Golden Search algorithm, but it seems that you are overwritting fx by accident. If I remove those line, it gives a more plausible result:

%%%Author: Jacob Joshua Shila
%%%Golden Search Algorithm
clear all
clc 

syms x
%%Input
fx = 0.4/sqrt(1+x^2)-sqrt(1+x^2)*(1-.4/(1+x^2))+x;
maxit = 50;
es = 10^-5;
R = (5^.5-1)/2;
%%Determine the Interval for the Initial Guess
x=[-10:10];
f = subs(fx,x);

xlow = 0.5;
xhigh = 1.5;
%%Perform Golden Search
xl = xlow;
xu = xhigh;
iter = 1;
d = R*(xu-xl);
x1 = xl+d;
x2 = xu-d;
f1 = subs(fx,x1);
f2 = subs(fx,x2);
if f1>f2
    xopt = x1;
else
    xopt = x2;
end
while(1)
    d = R*d;
    if f1>f2
        xl = x2;
        x2 = x1;
        x1 = xl+d;
        f2 = f1;
        f1 = subs(fx,x1);
    else
        xu = x1;
        x1 = x2;
        x2 = xu-d;
        f1 = f2;
        f2 = subs(fx,x2);
    end
    iter = iter+1;
    if f1>f2
        xopt = x1;
    else
        xopt = x2;
    end
    if xopt~=0
        ea = (1-R)*abs((xu-xl)/xopt)*100;
    end
    if ea<=es||iter>=maxit,break
    end
end
Gold = xopt

Returns:

Gold =

      1.0519

I believe this is the correct result. You get the same result if you take the derivative and search for its roots:

>> double(solve(diff(fx), 'Real', true))

ans =

      1.0519

[Jacob]

Thank for the input, it now works fine.

[ilyada]

How can we create a plot for this function?

Answer 2

[Aslain]

Hello,

I am looking for both the minimum of a function and the value at which this minimum is reached by the Golden Search method.

I need you to help me with the complete code directly implementable on matlab.

The function in question is the following with its various parameters.

f(x)=(((x.^(beta-1))*cc*A0)/(k*(nu^beta)))+(A2./(x*T))

cc=150000;

cp=800000;

cov=6000000;

k=5;

alfa=0.8;

beta=3.3;

nu=2100;

A0=1;

for j=1:k-1

A0=A0+exp(j*alfa);

end

A2=(k-1)*cp+cov.

Additional data for this function is as follows:

a=0; % start of interval

b=2; % end of interval

epsilon=0.000001; % accuracy value

iter= 100; % maximum number of iterations

r = (sqrt(5) - 1)*0.5; % golden proportion coefficient, around 0.618

I really ask you to help me, because I spent several sleepless nights without finding solutions.

[Walter Roberson]

https://www.mathworks.com/matlabcentral/answers/34570-golden-search-optimization-technique#answer_43383 but reverse the > to < because you are looking for minimum

Answer 3

[Aslain]

I followed your instructions to the letter, but unfortunately it still does not give me the correct minimal solution and the exact postion of this solution.

Indeed by direct derivation, the minimum obtained is indeed 1460.5 obtained at the position of x=1807.6 .

When I implement the code proposed for my case for minimization (respecting the change of inequalities that you had proposed to me), I obtain as minimum value of the function, the value 1.9534e+003 , obtained for x= 999.9999 .

There is indeed a strong disparity around the optimal solution (given by the direct derivation).

Can you offer me the correct full code?

The code I implemented for my case is as follows:

%%%Golden Search Algorithm

clear all

clc

syms x

%%Input

cc=150000;

cp=800000;

cov=6000000;

k=5;

alfa=0.8;

beta=3.3;

nu=2100;

A0=1;

for j=1:k-1

A0=A0+exp(j*alfa);

end

A2=(k-1)*cp+cov;

fx=(((x.^(beta-1))*cc*A0)/(k*(nu^beta)))+(A2./(k*x));

maxit = 100;

es = 10^-5;

R = (5^.5-1)/2;

%%Determine the Interval for the Initial Guess

x=[0:5000];

f = subs(fx,x);

xlow = 100;

xhigh = 1000;

%%Perform Golden Search

xl = xlow;

xu = xhigh;

iter = 1;

d = R*(xu-xl);

x1 = xl+d;

x2 = xu-d;

f1 = subs(fx,x1);

f2 = subs(fx,x2);

if f1<f2

xopt = x1;

else

xopt = x2;

end

while(1)

d = R*d;

if f1<f2

xl = x2;

x2 = x1;

x1 = xl+d;

f2 = f1;

f1 = subs(fx,x1);

else

xu = x1;

x1 = x2;

x2 = xu-d;

f1 = f2;

f2 = subs(fx,x2);

end

iter = iter+1;

if f1<f2

xopt = x1;

else

xopt = x2;

end

if xopt~=0

ea = (1-R)*abs((xu-xl)/xopt)*100;

end

if ea<=es||iter>=maxit,break

end

end

Gold = xopt

Answer 4

[Aslain]

I apologize, actually I was wrong. I set xhigh=1000 instead of xhigh=5000.

Your code works fine.

Thank you very much for your help, God bless you.

Answer 5

[Aslain]

I am dealing with a maintenance problem (reliability calculation...) according to the weibull model, to draw the point cloud

I consider the sample X of the TBFs, (time between failures)

X=[ ]

wblplot(X)

and it gives me the scatter plot on the logarithmic weibull plot

the problem is how to have the parameters of weibull, (beta, gamma and bare) on MATLAB, with which command, function??

Thank you for your help with an example please