Suppose we have a finite set of generators for an ideal $I \subset R := \Bbbk[x_1,\dotsc, x_n]$, where $\Bbbk$ is a field. If we choose a monomial ordering, then Buchberger's algorithm allows us to produce a Groebner basis for $I$. However, the size of the resulting Groebner basis can be enormous, and moreover can vary greatly depending on the monomial order chosen.

Sometimes, we have a reason for desiring a monomial order independent of $I$. (E.g., for elimination, we need an order with certain characteristics of lex; if we want to see of two sets of generators give the same ideal, we obviously want to use the same monomial order for both of them.) However, there are times when we may want to find a Groebner basis for $I$ with respect to *some* monomial, and we don't really care which. This could be useful, for instance, if we want to find a monomial $\Bbbk$-basis for $R/I$, and thereby (assuming $I$ is homogeneous) calculate the Hilbert polynomial of $I$.

Are there studies of algorithms and/or heuristics that design a monomial order based on the given generators of $I$ in an effort to produce a smaller Groebner basis for this particular ideal?

Ideally, it might be possible to choose a monomial order that has a good chance of outperforming grevlex on this particular generating set. At the very least, there should be some sort of heuristics for which grevlex order to choose (i.e., how the variables should be ordered).