(sorry for the terrible title, I don't know if there is a proper name for a function like that)
I have a function of several (n~100) real variables $f(x_1, x_2, \ldots, x_n)$ and I want to find the minimum. I know that the function has the following structure:
$f(x_1, x_2, \ldots, x_n) = \prod_{i\geq j} f_{ij}(x_i, x_j) = f_{11}(x_1) \times f_{12}(x_1, x_2) \times f_{13}(x_1, x_3) \times \cdots \times f_{22}(x_2) \times f_{23}(x_2, x_3) \times \cdots$
where all the combinations of $(1,\ldots, n)$ are used, including the repeated ones (e.g. $(1,1)$).
$f_{ij}$ functions have good properties, they are continuous and vary quite slowly. The minimum of $f$ is not very different from the minimum of $f_{11}$, $f_{12}$, ... but of course not identical (actually they are likelihood). Functions on the diagonal ($f_{11}$, $f_{22}$, ...) should have a sharpest minimum.
Computing each of $f_{ij}$ is very expensive, so it would be good if the optimization algorithm can search the minimum of $f$ changing only few (or one) parameters for each iteration, so that I have to recompute only few $f_{i,j}$ at each iteration. I don't know the gradient of $f$.
I am wondering if these kind of functions have been already studied and if there are suitable optimization algorithms.