Maximising the summation of a matrix elements along to generated trajectory by nonlinear global optimization

Hello every one,

I have a known matrix with n*m size that named fij.

I sum elements of this matrix along a trajectory (like below figure, blue element=1 and white element=0, the summation along red curve trajectory is equal to 10). I calculate this trajectory with a nonlinear function named t, which is a function of three unknown parameters (t(x1,x2,x3)). This function generates a 1*m vector.

I am going to find these 3 unknown parameters that generate a trajectory (or row index of elements for each column). t(j) is the row index of an element in the jth column. If I sum up these elements I will reach maximized value. My objective is that maximize the summation of fij elements along to generated trajectory (or (t(j),j) index) by nonlinear global optimization. If it is possible, please help me that how can I do this in Matlab. Please assume that we don't want to use error function such as least squares.

fij=[f(1,1) ... f(1,m)

.

.

.

f(n,1) ... f(n,m)], size (fij) = n*m

tm=[t(1) ... t(m)], size (tm) = 1*m

Maximze ---> sum(fij (t(j),j)), for j=1:m

Many thanks for your considerations in advance.

Best regards

Answer 1

[Matt J]

If fij is always binary, it seems very peculiar not to pose this as a least squares curvefit.

If f is not binary (or even if it is), you could apply ga, in which case I would probably write the fitness function like this:

x=ga( @(x)objective(x,f)  ,  ________________  );

function val = objective(x,f)

   [n,m]=size(f);
   
   tj=t(x);
   j=1:m;

   val = -sum(  interp2(f,j,tj) );

 end