Particle Swarm Optimization Toolbox for Non-linear Equation System Solving

Dear Matlab users,

I would like to use Particle Swarm Optimization Toolbox for Non-linear Equation System Solving. I got some errors. What I am doing wrong here?

The 3 non-linear equations are as follows:

Therefore, ,,will be minimized. If we describe , we can also write will be minimized. Therefore, In Particle Swarm Optimization (PSO), I thought to minimize the epsilon ϵ to be minimize near to zero. Then the 3 non-linear equation would be solved numerically. are coefficients.

My PSO Matlab code is as follows:

G1=2543309.32;H1=2211400.81;I1=3699427.32;
G2=446541.81;H2=1040240.96;I2=1167240.32;
G3=-5409318.49;H3=-4301537.69;I3=-5103577.6;
G4=953461.98;H4=194392.77;I4=675534.86;
G5=1561353.72;H5=1174001.77;I5=695136.68;
fun = @(x)((G1*x.^2+G2).*x(1).^2+(G3*x).*x(1)+(G4*x.^2+G5)).^2 + ((H1*x(1).^2+H2).*x(2).^2+(H3*x(1)).*x(2)+(H4*x(1).^2+H5)).^2+((I1*x.^2+I4).*x(2).^2+(I3*x).*x(2)+(I2*x.^2+I5)).^2

fun = function_handle with value:
    @(x)((G1*x.^2+G2).*x(1).^2+(G3*x).*x(1)+(G4*x.^2+G5)).^2+((H1*x(1).^2+H2).*x(2).^2+(H3*x(1)).*x(2)+(H4*x(1).^2+H5)).^2+((I1*x.^2+I4).*x(2).^2+(I3*x).*x(2)+(I2*x.^2+I5)).^2

lb = [0,0,0];
ub = [6,6,6];
options = optimoptions('particleswarm','SwarmSize',100,'HybridFcn',@fmincon);
rng default  
nvars = 3;
x = particleswarm(fun,nvars,lb,ub,options)

Error using particleswarm>makeState
Objective function must return scalar values.

Error in particleswarm>pswcore (line 177)
state = makeState(nvars,lbMatrix,ubMatrix,objFcn,options);

Error in particleswarm (line 154)
[x,fval,exitFlag,output] = pswcore(objFcn,nvars,lbRow,ubRow,output,options);

In my code above, I used x as λ, as μ, as ν.

Initial conditions(upper and lower limits for ) are between: .

I used the PSO example here on MATLAB website and edited the codes.

Could you help me with the code? Where is my wrong here? Thank you in advance.

[Torsten]

Your function "fun" returns a 3x1 vector, not a scalar value as required.

The reason is that there are some operations with the complete vector x, not only with its components.

Answer 1

[Bala Tripura Bodapati]

Hi Ercan

It is my understanding that the following error is encountered when you are trying to solve the system of non-linear equations using Particle Swarm Optimization:

'Objective function must return scalar value'

As per the documentation, the objective function should return a scalar value. In the following line of code specified, the entire vector 'x' is used in some parts of the equation, which results in the function returning a vector of 3x1 instead of a scalar value:

fun = @(x)((G1*x.^2+G2).*x(1).^2+(G3*x).*x(1)+(G4*x.^2+G5)).^2 + ((H1*x(1).^2+H2).*x(2).^2+(H3*x(1)).*x(2)+(H4*x(1).^2+H5)).^2+((I1*x.^2+I4).*x(2).^2+(I3*x).*x(2)+(I2*x.^2+I5)).^2

A possible workaround is to eliminate the use of entire vector in the objective function or assign a scalar value to 'lambda' values in the system of linear equations.

Refer the Particle Swarm Optimization documentation for more information.