optimization - Octave minimization for a many-body Hamiltonian with non-linear constraint

Question

I work in theoretical physics, and I have come upon a problem that requires the minimization of a particular Hamiltonian operator for a system of 8 particles, with one non-linear constraint. Due to the complexity of the system, I cannot define the entire Hamiltonian "in one go", nor the constraint. By this I mean that the quantity I am searching for is defined recurrently, depending on complex summations over quantities calculated for systems of 7 particles, which in turn depend on quantities calculated for systems of 6, and so on, until it reaches a one or two-particle system, for which said quantities are given as initial values, dependent on the elements of a column vector (the argument/minization parameters). The constraint itself is also of this form, requiring the "overlap" between the states of 8 particles to be exactly 1. (I.E. the state be normalized) I have been thinking of a way to use fmincon for this, but I've come up short, since my function has an implicit dependence on the parameters, and I can't write the whole thing explicitly. For a better understanding, here is some of the code:

for m=3:npairs+1
  for n=3:npairs+1
    for i=1:nsps
      for j=1:nsps
        overlap(m,n)=overlap(m,n)+x(i)*x(j)*(delta(i,j)*(overlap(m-1,n-1)-N(m-1,n-1,i))+p0p(m-1,n-1,j,i));
        p(m,n,i)=(n-1)*x(i)*overlap(m,n-1)-(n-2)*(n-1)*x(i)*x(i)*((m-1)*x(i)*overlap(m-1,n-1)-(m-2)*(m-1)*x(i)*x(i)*p(m-1,n-1,i));
        N(m,n,i)=2*(n-1)*x(i)*p(n-1,m,i);
        p0p(m,n,i,j)=(m-1)*(n-1)*x(i)*x(j)*overlap(m-1,n-1)-(m-1)*(n-1)*(m-2)*x(i)*x(i)*x(j)*p(m-2,n-1,i)-(m-1)*(n-1)*(n-2)*x(i)*x(j)*x(j)*p0(m-1,n-2,j)-(m-1)*(n-1)*(m-2)*(n-2)*x(i)*x(i)*x(j)*x(j)*(delta(i,j)*(overlap(m-2,n-2)-N(m-2,n-2,i))+p0p(m-2,n-2,j,i));
      endfor
    endfor
  endfor
endfor

function [E]=H(x)
    E=summation over all i and j of N and p0p for m=n=8 %not actual code


endfunction
overlap(9,9)=1 %constraint

question from:https://stackoverflow.com/questions/65876443/octave-minimization-for-a-many-body-hamiltonian-with-non-linear-constraint

Answer 1

It's hard to give a specific answer, but I would advise the following to get you started.

First, note that, the inner two steps of the nest loop can be vectorised, since i and j always appear as indices (whereas m and n make backreferences, so they cannot be vectorised). So your 4-level loop can be reduced to a 2-level loop containing 4 functions operating over i-by-j matrices.

Second, note that the whole construct can be expressed as a recursive function. If you have suitable base cases for m = 0, n = 0, you can iteratively obtain all i,j matrices for all cases up to m=9,n=9. In particular, you can try to 'memoize' the early steps, and plug them into higher steps, rather than rely on actual recursion.