# Polynomial time optimization problems belong to which complexity class?

I know that $$\mathsf{P}$$ class is only defined for decision problems. Therefore, a problem like "Does there exist an $$(s,t)$$ path of length $$k$$ in the graph $$G$$?" is in $$\mathsf{P}$$. One can first find the shortest path from $$s$$ to $$t$$ in polynomial time and check if it is smaller than $$k$$ or not.

Now, consider the optimization version of this problem that we popularly call the shortest path problem: "Find a shortest $$(s,t)$$ path in the graph $$G$$". Since this is not a decision problem, is it wrong to say that "shortest path problem is in $$\mathsf{P}$$"? If so, in which complexity class does it belong to?

Technically speaking, it is wrong. However, usually people don't make a huge difference between the two. Its usually clear from the context whether the problem is a decision or a search problem, hence its clear whether it should be in $$P$$ or $$FP$$.
So, writing that "Finding the shortest $$(s,t)$$ path in the graph $$G$$, is a $$P$$ problem" would actually mean: "Finding the shortest $$(s,t)$$ path in the graph $$G$$, is an $$FP$$ problem" (since the context is clear)