Multi Objective nonlinear functions with unknown multi parameters

Dear Matlabers,

Lets say my system of equations are as below:

((1-x(1))*(10+x(5)))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.154;

((1-x(1))*(10-x(5)))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.254;

(x(1)*(10+x(4)))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.545;

(x(1)*(10-x(4)))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.034;

(10+(x(1)*x(4))+(1-x(1))*x(5))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.994;

(10-(x(1)*x(4))-(1-x(1))*x(5))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.063;

and I want to find x(1) ... x(5) values so that the equalities hold.

I already tried so many optimization methods but the problem is, after finding the values when I replace them with x values the error values I came across are very large.

Any help is appreciated..

Best,

[Torsten]

Where is x(6) in your equations ?

[SN MJTHD]

You are right my fault. I correct it to x(1) ... x(5).

[Torsten]

Then your system is overdetermined and errors in the equations for a least-squares solution from an optimizer are normal.

[SN MJTHD]

Yes that exactly is and I am trying to find the best way with the minimum error.

[Torsten]

lsqnonlin, e.g., will give you a solution for x(1),...,x(5) that minimizes the sum of errors squared of the equations. Do you have another error in mind ?

[SN MJTHD]

I want to solve the whole equations simultaneously but the error for each equation be minimum not the square root of sum of squared errors of each equation.

I mean when I calculate the x(1) ... x(5) values and replace them for example in any of the

((1-x(1))*(10+x(5)))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.154;

or

(10-(x(1)*x(4))-(1-x(1))*x(5))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.063;

equations, the equality holds.

I already tried many methods like fmincon, fsolve, gamultiobj but I was not satisfied by the results,

[Torsten]

I mean when I calculate the x(1) ... x(5) values and replace them for example in any of the

((1-x(1))*(10+x(5)))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.154;

or

(10-(x(1)*x(4))-(1-x(1))*x(5))^2/(((x(1)^2)*x(3)^2)+((1-x(1))^2*x(2)^2)) = 0.063;

equations, the equality holds.

Say (for simplicity) your equations read

x(1) = 7

x(1) = 13

(2 equations in one unknown), you can't simultaneously find x(1) that makes the error in both equations equal to zero. The best compromise is x(1) = (7+13)/2 = 10.

That's the problem if you have more equations than unknowns.

[SN MJTHD]

so in this case I think the best way is to use fmincon to minimize the square root of overall error.

[Torsten]

lsqnonlin is especially designed to solve problems like yours, but you are of course free in the choice of the optimizer.

[SN MJTHD]

Thank you for your time and your effort.

Best

Answer 1

[Alex Sha]

There are multi-solutions:

1:

x1: 0.387245311047365

x2: -0.00972723068513297

x3: -40.2902696169456

x4: 18.9579782048437

x5: -2.65707955299356

2:

x1: 0.387245311627332

x2: 7.2906925850934

x3: 38.6033440479639

x4: 18.9579781524691

x5: -2.65707955385685

3:

x1: 0.387245311366753

x2: -24.1738267443381

x3: 12.6548907569997

x4: 18.9579781707223

x5: -2.65707955330091