Multi Objective nonlinear functions with unknown multi parameters

2019 7 22
1 个回答
更新时间：2019 8 28
1 次查看（30 天）

0 个投票

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,

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SN MJTHD
SN MJTHD 2019-7-23
You are right my fault. I correct it to x(1) ... x(5).
Torsten
Torsten 2019-7-23
Then your system is overdetermined and errors in the equations for a least-squares solution from an optimizer are normal.
SN MJTHD
SN MJTHD 2019-7-23
Yes that exactly is and I am trying to find the best way with the minimum error.
Torsten
Torsten 2019-7-23
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
SN MJTHD 2019-7-23
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
Torsten 2019-7-23
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
SN MJTHD 2019-7-23
so in this case I think the best way is to use fmincon to minimize the square root of overall error.
Torsten
Torsten 2019-7-23
lsqnonlin is especially designed to solve problems like yours, but you are of course free in the choice of the optimizer.
SN MJTHD
SN MJTHD 2019-7-23
Thank you for your time and your effort.
Best

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回答（1 个）

Alex Sha
Alex Sha 2019-8-28

0 个投票

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

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SN MJTHD
SN MJTHD
2019-7-22

回答：

Alex Sha
Alex Sha
2019-8-28

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