# Finding a global minimum

I seek the function $$f$$ which satisfies the 100 equations (i=1,2...100)

$$\sum_{j=1}^{2000} f(A_{ij},B_{ij},C_{ij})=Q_i$$.

Where $$A,B,C$$ are 100x2000 matrices and all entries are between 0 and 1.

So I know Q, A, B and C
How can one find the function?

I tried to parametrize the function $$f_w$$ using trilinear interpolation. And then I minimized the error function $$\Psi(w) = \sum_i \left(\left(\sum_j f_w(A_{ij},B_{ij},C_{ij})\right) - Q_i\right)^2.$$

I minimized by just walking around randomly until a minimum is found. But when I ran the simulation many times it finds different minimums. I want to find the smallest minimum, or at least a pretty good one. Is there any method for this? Would neural networks work better?

I have a second dataset to test the function.

If I understand these methods (such as neural networks and gradient descent) correctly they approximate the gradient at your current point, so it will just find a local minimum and wont work any better then my method that just walks a step in a random direction and keeps it if the error function decreases?

Are there any methods to find a good global minimum? What is the best method to find a function that fits this data with least error function?

• It actually depends on the parameterization you choose. What does it mean 'trilinear interpolation' in your case? What are the grid points? If you can write your function $f$ as a linear combination of basis functions, then you get a simple linear least-squares problem. – Alex Shtof Mar 4 at 11:45
• @AlexShtof I used a 5x5x5 cubic grid and interpolated between lattice values – KALLE DA BAWS Mar 4 at 11:50
• @AlexShtof Can you explain this method a bit more, and why you think it will work ? – KALLE DA BAWS Mar 4 at 11:51

For solving this kind of problem, one cannot avoid choosing a parametrisation. You may choose it yourself like you have done above, or you may write something more general like a neural network (which is just a very nonlinear parametrisation). However the choice matters and you are likely to find different minima when you vary it. Do you have any intuition about the data? Do you expect $$f$$ should have a certain property? This might help making the right choice.