How do I optimize this three variable quadratic equation to minimize one of the variables?

I have a system which can be modeled by the equation

I'm trying to find the smallest value of mu for varying values of x and L such that the equation above holds true.

The values for x and L are constrained as follows:

Answer 1

[Ameer Hamza]

Try this

% minimize abs(mu) <=> minimize mu^2
% X(1)=x, x(2)=L, x(3)=mu

A = [-0.5 1 0];
B = 0;
x_sol = fmincon(@(X) abs(X(3)), rand(1,3), A, B, [], [], [1 0 0], [15 inf inf], @nlcon);
mu_sol = x_sol(3);

function [c, ceq] = nlcon(X)
    x = X(1);
    L = X(2);
    mu = X(3);
    ceq = (0.5 + x+2*L)*mu^2 + 0.5*mu - 2*L;
    c = [];
end

The minimum value of mu turns out to be very small

mu_sol =
   5.1200e-07

[Zachary Fisher]

Okay, I think I'm following.

I had tried fmincon, but for the "fun" term I tried to use my original equation. Why did you use abs(mu)? Is it because what I'm trying to minimize is mu, and mu must be positive?

You also describe this as minimizing mu^2, is that a typo?

Other than that, it seems to work based on what I expected. Thanks for setting me in the right direction.

[Ameer Hamza]

yes, that was a typo. I initially tried mu^2, and then I tried abs(mu), because minimizing both will give an optimal solution. You can use either.

I used abs(mu) as an objective function because the goal is to minimize its value. Whereas, the equation is a constraint on the type of solution you want.

[Zachary Fisher]

I was trying to increase the complexity a bit and I'm running into issues and am hoping you can help me some more.

My original equation, which you defined as ceq, was based off of three equations of motion that were simplified down to one.

Where D is set constant at the beginning of the problem (For my original case I had set it to 0.5). I then took these three equations of motion and simplified and rearranged them down to the original one equation. All of the Nrw, Nfw and W happened to cancel out. For a more complex system involving more complex variables or more equations, I don't want to have to do that by hand, and it might not simplify so easily.

So my question is, is there a way to set up fmincon with multiple ceq equations, and some variables that are preset at the beginning, like D and W in this case? For those three equations Nrw and Nfw are dependent on the geometry, so I would still only want MATLAB to adjust x and L such that mu is minimized.

Is this possible with fmincon?

So like, ceq would be equal to all three of those equations, D = 0.5, W = 100, etc.

[Ameer Hamza]

Yes, you can setup the nlcon for multiple equality constraint like this

function [c, ceq] = nlcon(X)
    x = X(1);
    L = X(2);
    mu = X(3);
    
    D = 0.5; % known constants
    W = 100;
    
    ceq(1) = % write the first equation
    ceq(2) = % write the second equation
    ceq(3) = % write the thirs equation
    ceq = ceq(:); % convert to a column vector
    c = [];
end

[Md Muzakkir Quamar]

Hi, I have a similar problem. i want to minimize a function with multi variable and constraint. i used fmin function but i think i am making some error.

here is the problem : Let P1 = (x1, y1) and P2 =

(x2, y2) be two given points. Find the third point P3 = (x3, y3) such that

d1 = d2 is minimized, where d1 is the distance from P3 to P1 and d2 is the

distance from P3 to P2.

so objective function is d1 + d2 =0 and constraint is d1=d2

where d1 = [(x3-x1)^2 +(y3-y1)^2]^1/2 & d2= [(x3-x2)^2 +(y3-y2)^2]^1/2

can you please help

[BraF]

Hello Ameer,

Could you tell me why in the code above, you wrote the second value of 'ub' ([15 inf inf]) as infinity ? The constraint on L is that it has to be between 0 and 0.5x, so could you explain me how to incorporate this (constraints that depend on one of the variables and that are not bounded between two scalars) ?

In my case, I have the following constraints for example :

1<x1<2 ;

x1/10<x2<3x1 ;

...

Thanks in advance!