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Unable to find explicit solution in Lagrangian optimization
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I am trying to find the analytical solution to the following problem:
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1614056/image.png)
I tried solving it by coding the Lagrangian by hand and use solve, but Matlab prints the warning: "Unable to find explicit solution".
I used the following code:
syms e1 e2 p1 p2 rho gamma lambda
syms E H(e1,e2)
H(e1,e2) = (e1^rho +e2^rho)^(1/rho)
L(e1, e2, lambda) = p1*e1 +p2*e2 + lambda*(H(e1,e2)-E)
L_e1 = diff(L,e1) == 0
L_e2 = diff(L,e2) == 0
L_lambda = diff(L,lambda) == 0
system = [L_e1,L_e2,L_lambda]
[e1_s,e2_s,lambda_s]=solve(system,[e1 e2 lambda])
Do you know what I could do to solve this? Or is there a different and better way to find an analytical solution?
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采纳的回答
Catalytic
2024-2-11
编辑:Catalytic
2024-2-11
An analytical solution for 0<rho<1 is -
A=[1 0;
0 1;
-1 0;
0 -1]*E;
[fval,i]=min(A*[p1;p2]);
e1=A(i,1);
e2=A(i,2);
2 个评论
Matt J
2024-2-11
I like it. And, in fact, because the extreme points lie at points where H(e1,e2) is not differentiable, it shows that you will never find the true solution with Lagrange multiplier analysis.
更多回答(1 个)
Matt J
2024-2-11
If you make rho explicit, it seems to be able to find solutions. I doubt there would be a closed-form solution for general rho.
rho=2;
syms e1 e2 p1 p2 gamma lambda
syms E H(e1,e2)
H(e1,e2) = (e1^rho +e2^rho)
L(e1, e2, lambda) = p1*e1 +p2*e2 + lambda*(H(e1,e2)-E^rho)
L_e1 = diff(L,e1) == 0
L_e2 = diff(L,e2) == 0
L_lambda = diff(L,lambda) == 0
system = [L_e1,L_e2,L_lambda]
[e1_s,e2_s,lambda_s]=solve(system,[e1 e2 lambda])
4 个评论
Matt J
2024-2-11
Even when it can be explicitly solved, the result isn't nice:
rho=sym(1/4);
syms e1 e2 p1 p2 gamma lambda
syms H(e1,e2)
H(e1,e2) = (e1^rho +e2^rho);
L(e1, e2, lambda) = p1*e1 +p2*e2 + lambda*(H(e1,e2)-1);
L_e1 = diff(L,e1) == 0;
L_e2 = diff(L,e2) == 0;
L_lambda = diff(L,lambda) == 0;
system = [L_e1,L_e2,L_lambda];
[e1_s,e2_s,lambda_s]=solve(system,[e1 e2 lambda])
Walter Roberson
2024-2-11
You can eliminate the root() constructs, but the result is confusing.
rho=sym(1/4);
syms e1 e2 p1 p2 gamma lambda
syms H(e1,e2)
H(e1,e2) = (e1^rho +e2^rho);
L(e1, e2, lambda) = p1*e1 +p2*e2 + lambda*(H(e1,e2)-1);
L_e1 = diff(L,e1) == 0;
L_e2 = diff(L,e2) == 0;
L_lambda = diff(L,lambda) == 0;
system = [L_e1,L_e2,L_lambda];
[e1_s,e2_s,lambda_s]=solve(system,[e1 e2 lambda], 'maxdegree', 3)
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