It’s a classical point—you can replace the question of “What is the minimal value of f(x) (what is the witness x that gives the minimal value)?” by “Is there a parameter x that gives the value f(x) less than C (what is an example of such x)?”, and then use binary search on C to pinpoint the minimum. Since being able to write down the answer in polynomial time must be a given, you can take the ability to run the search on C in polynomial time for granted.
It’s a classical point—you can replace the question of “What is the minimal value of f(x) (what is the witness x that gives the minimal value)?” by “Is there a parameter x that gives the value f(x) less than C (what is an example of such x)?”, and then use binary search on C to pinpoint the minimum. Since being able to write down the answer in polynomial time must be a given, you can take the ability to run the search on C in polynomial time for granted.