In this context we can divide problems into two cases: optimization problems ("find a solution that satisfies P, in a way that has the highest score") and decision problems ("is there a solution that satisfies P, with score >= X" or "find a solution that satisfies P, with score >= X").

Complexity classes like NP are defined only for decision problems, not for optimization problems. NP can be defined either as

* the set of decision problems where, given a solution, you can check it in polynomial time with a deterministic Turing machine, or * the set of decision problems solvable in polynomial type by a non-determistic Turing machine

these two definitions being equivalent.

When someone mentions an optimization problem being in P, NP, or NP-hard, they actually mean the associated decision problem being in P/NP/NP-hard.

While technically incorrect, this is fine when working informally, because you can "translate" between the two in polynomial time.

If I give you an oracle (ie. a magic box) that solves an optimization problem (ie. gives you a solution to P with the highest score) immediately, you can trivially write a polynomial time algorithm that solves the decision problem: call the oracle, check the score of its answer, then compare that with the X value you were given. And vice versa: if I give you an oracle that solves a decision problem immediately (ie. given a value X, gives you a solution satisfying P with score >= X), you can write a polynomial time algorithm that uses the oracle a few times with the right values of X (exponential search then bisection), to find the solution with the highest score.