How to use augmented Lagrange multipliers for inequality constrains to solve min max problem with iterations

Dear all,

I have this lagrange equation with some inequalities constrains. I want to apply augmented Lagrange multipliers for inequality constrains to solve min max problem and get the optimized values of the lagrange multipliers(k_and v_) and the remaining variables if its possible

Please can anyone help me ?

I apprecitae your help.

Here is my code for the lagrange equation and constraints.

note that

subject to

here is my code of the equation and constraints.

TN=10;

expo=1;

pL=zeros(1,TN);

for l=1:TN

pL(l)=l^-expo;

end

beta=pL./sum(pL);

meu=pL./sum(pL);

zm=rand(1,10);

ro=rand(10,10); %

x=zm*ro;

CMRN=0.4*(10^5);

aa = 0.1;

bb = 1;

Fkf= ((bb-aa).*rand(10,1) + aa)*(10^9);

Rk=1.0e+08 *[ 3.0346, 4.2479,5.8977,5.7046, 6.1621,5.8498,5.7329,6.0979, 5.8282,6.2186];

a = 500;

b = 2000;

Lks = ((b-a).*rand(10) + a);

CK=[500,600,700,800,900,1000,1100,1200,1300,1400];

ls=[20000,20000,20000,20000,20000,20000,20000,20000,20000,20000];

for k=1:10

for s=1:10

(zm(s)*ls(s))<=CMRN; % the constraint (11f)

equation(k,s)=beta(k)*((Lks(k,s)*meu(s))/Rk(k)+(CK(k)*Lks(k,s)*meu(s))/Fkf(k))*x(k,s)+beta(k)*((Lks(k,s)*meu(s))/Rk(k)+(CK(k)*Lks(k,s)*meu(s))/Fkf(k))*(1-zm(s))+k_*(x(k,s)-zm(s))+v_*(x(k,s)-ro(k,s))

end

end

lagrange_equation=sum(sum(equation));

[Torsten]

If you read further in the article, you'll notice that the Lagrange function itself is never used in the optimization.

Focus on the three subproblems defined thereafter.

[bassant tolba]

Yeah.. you are right..I noticed that the partial derivatives for ro and zm and x are done. However, I cannot figure out the value of Lagrange multipliers(kaba ka and meu). Please do you have any idea how can I calculate with some iteration.

Thank you alot Torsten

[Torsten]

(39) - (42) show how kappa and nu are updated after solving the three subproblems for rho, z and x.

[bassant tolba]

Yeah.. I calculated the optimum ro from equation 38 And then I put kappa and nu equal zeros in equation 36 and 37 ( as an initial guess ) and got the initial optimum values for z and x Can you kindly help me in writing the part of updating the kappa and mu according to equation 39 and 40. I will be grateful for your help.

[Torsten]

And what is not clear in 39 and 40 ? The step size ? I think it's some positive value between 0 and 1. The smaller you choose it, the longer it takes to achieve convergence. The bigger you choose it, the bigger the risk to diverge. Or do you know better from another part of the article or from the thematic background what "step size" could mean ?

[bassant tolba]

Yeah.. I’m sorry if I’m asking, but can you kindly share with me the MATLAB code of this updating with some iterations... I can not imagine its MATLAB code. Thanks in advance.

[Torsten]

I’m sorry if I’m asking, but can you kindly share with me the MATLAB code of this updating with some iterations

Sorry - no. This is overcharged.

[bassant tolba]

It’s obvious that anyone who are asking in Mathworks that they are not masters of MATLAB language and they need help for solving problems. Thus, Mathworks is established for sharing the knowledge and help each other not for refusing and considering a question as a trivial like you considered from your side. Anyway. Thanks!

[Torsten]

...considering a question as a trivial like you considered from your side.

At the contrary: I consider your question as much too difficult for me to answer since I still don't understand the model and I don't have the necessary background knowledge for coding it. So I had to invest a large amount of time that I'm not willing to spend on this. Your questions don't mainly address MATLAB skills, but understanding of the underlying subject. What I answered so far were just guesses on how I understood some passages of the article.

And besides this, all participants in this forum are volunteers - so you can ask for help, but voluntary help has limits.