Karp reduction from optimization problems to decision problems

Question

When you consider Cook reductions, then decision and optimization versions of the problems are polynomial time reducible to each other.

Focusing on Cook reductions, there exists a natural Karp reduction from the decision version of a problem to optimization version. Is the converse also true?

Answer 1

Thanks to comments by Yuval Filmus, I understand that my question does not make sense as Karp reductions are defined for decision problems. Since Cook reductions allow more freeness, it makes sense to talk about a Cook reduction from a decision problem to an optimization problem, but this is not true for Karp reduction.