A polynomial-time algorithm was presented in this paper:
Chew, L. Paul, Michael T. Goodrich, Daniel P. Huttenlocher, Klara Kedem, Jon M. Kleinberg, and Dina Kravets. "Geometric pattern matching under Euclidean motion." Computational Geometry 7, no. 1-2 (1997): 113-124.
Elsevier journal link.
If I read their results correctly,
"the minimum (bidirectional) Hausdorff
problem under Euclidean motion" for polygons
can be solved in time $O((m + n)^6 \log^2 mn)$, for polygons of $n$ and $m$ vertices.
They did not specialize to convex polygons.
These are primarily theoretical results, and I doubt have been implemented.
Other work focuses on approximation algorithms, e.g.,
Alt, Helmut, Bernd Behrends, and Johannes Blömer. "Approximate matching of polygonal shapes." Annals of Mathematics and Artificial Intelligence 13, no. 3-4 (1995): 251-265. Springer link.
A more practical algorithm specialized to convex polygons is described here:
Danilov, Dmitry I., and Aleksei Stanislavovich Lakhtin. "Optimization of the algorithm for determining the Hausdorff distance for convex polygons." Ural Mathematical Journal 4, no. 1 (2018): 14-23.