How can I solve an Optimization problem?

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Juan Barrientos 2021-12-13

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评论： Torsten 2021-12-14

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)));

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Abolfazl Chaman Motlagh 2021-12-13

编辑：Abolfazl Chaman Motlagh 2021-12-13

does lambda has any bound ? like an interval? because acosd hence Mu_1 become imaginary in larg numbers.

can you provide simple value for d ?

Juan Barrientos 2021-12-13

Yes, d is the input. So is fi and psi. And the interval is between 0 and 1 for lambda.

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Answer 1

Abolfazl Chaman Motlagh 2021-12-13

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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)

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Abolfazl Chaman Motlagh 2021-12-13

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Yes it is. but are you sure you wrote the equations right? because it seems it is not what you're saying. lets plot the function over lambda:

d = 100;

psi = 30;

fi = 170;

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)))));

x = 0:1e-5:1;

for i=1:numel(x)

y(i) = Mu_1(x(i));

end

plot(x,y)

Warning: Imaginary parts of complex X and/or Y arguments ignored.

Juan Barrientos 2021-12-13

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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Answer 2

Juan Barrientos 2021-12-13

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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.

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Juan Barrientos 2021-12-13

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Yes maximize, but I said minimize because I think the software minimizes, so to maximize you have to multiply by (-1); that was my mistake But I already solve it. I used the function fminsearch to find the value for λ; I show you the code I used:

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_2= @(lambda) -(acosd(abs(((c(lambda)^2)+(b(lambda)^2)-((d+a(lambda))^2))/(2*b(lambda)*c(lambda)))));
min_2= abs(Mu_2(abs(lambdaMin2)));

I had to use Abs to solve for the maximum. Thank you very much for your responses, they helped me a lot.

Torsten 2021-12-14

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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