How can I solve an Optimization problem?

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Juan Barrientos
Juan Barrientos on 13 Dec 2021
Commented: Torsten on 14 Dec 2021
Hello. I have not used the optimization toolbox and I need your help. I have 3 functions that depends on λ, and an function μ that depends on the 3 previous functions (so μ also depends on λ). I need to find the minimum value of μ changing λ: how can I make it? what function should I consider? Thanks in advance.
syms lambda;
c= sqrt((-(d^2)*(cosd(psi)-1))/(1+cosd(fi-psi)+(lambda^2)*(1-cosd(fi-psi))));
a= lambda*c;
b= sqrt(((d^2)*(lambda^2)*(cosd(fi)-1)-1-cosd(fi))/((lambda^2)*(cosd(fi-psi)-1)-1-cosd(fi-psi)));
Mu_1= acosd(abs(((c^2)+(b^2)-((d-a)^2))/(2*b*c)));
  2 Comments
Juan Barrientos
Juan Barrientos on 13 Dec 2021
Yes, d is the input. So is fi and psi. And the interval is between 0 and 1 for lambda.

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Answers (2)

Abolfazl Chaman Motlagh
Abolfazl Chaman Motlagh on 13 Dec 2021
you can use fmincon, this function minimize function in a constraint problem. but only constraint here is bounds of lambda. so other fields of function are empty ([]).
i use some sample number for needed variables.
d = 1;
psi = rand * 360;
fi = rand * 360;
c=@(lambda) (sqrt((-(d^2)*(cosd(psi)-1))/(1+cosd(fi-psi)+(lambda^2)*(1-cosd(fi-psi)))));
a=@(lambda) (lambda*c(lambda));
b=@(lambda) (sqrt(((d^2)*(lambda^2)*(cosd(fi)-1)-1-cosd(fi))/((lambda^2)*(cosd(fi-psi)-1)-1-cosd(fi-psi))));
Mu_1=@(lambda) (acosd(abs(((c(lambda)^2)+(b(lambda)^2)-((d-a(lambda))^2))/(2*b(lambda)*c(lambda)))));
[Lambda_star,fval,exitflag,output]=fmincon(@(x) Mu_1(x),1,[],[],[],[],0,1);
Local minimum possible. Constraints satisfied. fmincon stopped because the size of the current step is less than the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
disp(Lambda_star)
0.1605
use fmincon documentation if you need more options for better convergence.it seems it reach best answer in my case : (in my code the answer changes everytime because psi and fi are random)
x = 0:1e-3:1;
for i=1:numel(x)
y(i) = Mu_1(x(i));
end
plot(x,y)
  3 Comments
Juan Barrientos
Juan Barrientos on 13 Dec 2021
Hmmm yes I'm sure, because I already know the solution of the problem, just don't know how to code. the solution is
c= 54.85
b= 75.34
a= 18.98
with lambda= 0.346 and if I solve the equations the answers in the program are these. But I don't really know how to use fmincon or quadprog for this.

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Juan Barrientos
Juan Barrientos on 13 Dec 2021
So to be clear these are the equations. φ, ψ and d are parameters, so the problem is to know which value of λ makes the smaller μ.
  3 Comments
Torsten
Torsten on 14 Dec 2021
Be careful with the objective function if the expression inside acosd becomes greater than 1. fminsearch will most probably stop if complex numbers are encountered during the optimization.

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