Hello! How can i solve optimization problem with MATLAB?

I want to optimize the following problem:

V=: x^2+y^2+z^2+w^2

Subject to: a*x^2+y^2+b*(z+p2)^2-D*(z+p2)*w+B*w^2-b*z=K

where a> 0, b>0, p2>0, D>0, B>0, K>0.

Can we solve this problem by using the Lagrange multiplication method in MATLAB?

Answer 1

[Ayush Aniket]

Hi Shadab,

As per my understanding you want to solve the optimization problem using Lagrange Multiplier method. You can solve it by defining the variables as symbolic and using MATLAB’s ‘solve’ function to solve the system of differential equations as shown below:

syms x y z w lambda 

% Define the function V and the constraint g
V = x^2 + y^2 + z^2 + w^2;
g = a*x^2 + y^2 + b*(z + p2)^2 - D*(z + p2)*w + B*w^2 - b*z - K;

% Define the Lagrangian
L = V - lambda * g;

% Compute the system of equations by taking the gradient of the Lagrangian
eqns = [
    diff(L, x) == 0,
    diff(L, y) == 0,
    diff(L, z) == 0,
    diff(L, w) == 0,
    diff(L, lambda) == 0
];

% Solve the system of equations
sol = solve(eqns, [x, y, z, w, lambda], 'Real', true);

However, the 'solve' function is used to find exact solution of system of normal or differential equations and cannot be used to find the maximum or minimum value of your objective function V. Please refer to the following documentation page to read more about the 'solve' function:

https://www.mathworks.com/help/symbolic/sym.solve.html

Also, since the derivative of the respective lagrangian functions are zero you may get maxima or minima based on solution you get after solving and will need to verify it. Another way to solve this problem would be to use 'fmincon' function which finds the minimum of the constrained optimization problem. Please refer to the following MATLAB answer for a detailed analysis:

https://www.mathworks.com/matlabcentral/answers/531298-finding-minimum-maximum-of-function-using-lagrange-multipliers

Hope it helps.