I agree with your premise that "optimize" is a transitive verb, but I don't think objects should necessarily be overt; they can be covert as in your example
The string has to do more things. It is harder to optimize
We call this a tough construction *, in which the subject in the main clause is the implied object in the infinitival clause. This contruction is permitted by tough adjectives—e.g easy, hard, difficult.
The passive counterpart is not possible since it doesn't take an object.
*Biber et al in the Longman Student Grammar of Spoken and Written English analyze it as an object-to-subject raising; I don't like this term, however, because it can be confusing to learners. Tough contruction is a good term because it reminds you of the tough adjectives.