How to optimize a linear system of complex-valued equations

Hello everyone

I have a linear system of complex-valued equations as follows

ax + by = c

dx + ey = f

where a,b,c,d,e, and f are complex-valeued coefficients and x and y are my complex-valued unkonwns. I intend to conduct an algorithm like genetic algorithm to optimize "a" and "e" to get the optimized (min for example) value of "y". Would you please help me how I can manage to do this? thanks in advance for your time devoted to this question.

[David Goodmanson]

Hi QD

by the 'minimum value of y' do you mean the minimum for abs(y)? If so, you set

a = dc/f, x = f/d and y = 0    [1]

then the two equations reduce to

c = c
f = f

So [1] is a solution that minimizes abs(y). No optimization with e is necessary,

[Proman]

Hi David

Thank you for your answer.

Well I am totally with your answer yet I am not supposed to use it but I may have been wrong in the way I asked my question. in the first place, you are absolutely right about the cost function as abs(y) but let me generalize my problem. my main problem is a system of 63 linear complex-valued equations and in anyways I can not follow this procedure to optimize it.

Let me put it this way:

Imagine that I have a system of 3 linear equations as follows:

a1x + a2y + a3z = a4

a5x + a6y + a7z = a8

a9x + a10y + a11z = a12

where a1 to a12 are complex cofficients and x, y and z are complex unknowns. Now I have a cost function (to be minimzed as) as:

cost = abs(i*C*(z + y))

where i is sqrt(-1) and C is a constant. for example, since y and z are the unkonws of interest in the system, how can I minimize cost function by adjusting a1, a6 and a12, to get appropriate complex values of z and y in a way that my cost function becomes as minimized as possible. I mean, I want to see for which image(a1) image(a6), image(a12), image(z), image(y), real(a1), real(a6), real(a12), real(z) and image(y) values, cost becomes minimum.

[Bruno Luong]

Trivial with your example:

a4, a8, a12 = 0,

Then x, y, z == 0 satisfied the linear system. Your cost is 0, the smallest possible.

[Proman]

Well that's absolutely correct but I have constraints for my decision variables. In other words, I intend to run the optimization process using an standard algorithm like genetic algorithm or PSO. In that way, how can I accomplish my optimization?

[Proman]

This is my main problem. Ro1 matrix is the coeffient maatrix which is 8*8. and it has 8 unknows. I want to optimize R1(4) + R1(7) to minimze cost function for 1<g3<2 and 1<Oc1<4.

%%%Part I ==> Constants Input
format long 

g2 = 3.5155;                   
g3 = g2;                       
g1 = 0;                                    
CP = 200;                      
Oc1 =  0;                      
k = 10000;                        


%%Part II => Bistability Relation, Im and Re part of Rho21 based on
% different values of Oc (D21=D32=0)

             
%%Preallocating Matrices

G21 = zeros(1,k);

G2 = zeros(1,k);

G3 = zeros(1,k);

G1 = zeros(1,k);

G31 = zeros(1,k);

G32 = zeros(1,k);

OC1 = zeros(1,k);

rho1 = zeros(1,k);

cost = zeros(1,k);

y2 = zeros(1,k);

y3 = zeros(1,k);

C = zeros(1,k);


wp = zeros(1,k);

    
    OP = linspace(0,50,k);
  


for j = 1 :k
    
  
    
    G2(1,j) = g2;
    
    G3(1,j) = g3;
    
    G1(1,j) = g1;
    
    G21(1,j) = (G1(1,j) + G2(1,j)) ./ 2;
    
    G32(1,j) = (G3(1,j) + G2(1,j)) ./ 2;
    
    G31(1,j) = (G3(1,j) + G1(1,j)) ./ 2;
    
    OC1(1,j) = Oc1;
    
    C(1,j) = CP;
    
 
    
    Ro1 = [-G2(1,j) -(1i*OP(1,j)) 0 (1i*OP(1,j)) 0 0 0 -G2(1,j);
        -(2i*OP(1,j)) (-1i*wp(1,j)-G21(1,j)) -(1i*OC1(1,j)) 0 0 0 0 -(1i*OP(1,j));
        0 -(1i*OC1(1,j)) (-1i*wp(1,j)-G31(1,j)) 0 (1i*OP(1,j)) 0 0 0;
        (2i*OP(1,j)) 0 0 (1i*wp(1,j)-G21(1,j)) 0 (1i*OC1(1,j)) 0 (1i*OP(1,j));
        (1i*OC1(1,j)) 0 (1i*OP(1,j)) 0 -G32(1,j) 0 0 (2i*OC1(1,j));
        0 0 0 (1i*OC1(1,j)) 0 (1i*wp(1,j)-G31(1,j)) -(1i*OP(1,j)) 0;
        -(1i*OC1(1,j)) 0 0 0 0 -(1i*OP(1,j)) -G32(1,j) -(2i*OC1(1,j));
        0 0 0 0 (1i*OC1(1,j)) 0 -(1i*OC1(1,j)) -G3(1,j)];
    
    B1 = [-G2(1,j);-(1i*OP(1,j));0;(1i*OP(1,j));(1i*OC1(1,j));0;-(1i*OC1(1,j));0];

    
    R1 = Ro1 \ B1;

    
    rho1(1,j) = R1(4) + R1(7);
 

    
    %input-output relation : |x| in terms of |y|
 
   cost(1,j) = (2 .* OP(1,j)) - (1i .* C(1,j) .* rho1(1,j));

  

end

%%%Part III ==> Plotting

figure
plot(abs(cost),OP)
xlabel('input |y|')
ylabel('output |x|')