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.

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Fig.5.
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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.
doi.