would complexity of constraint satisfaction problem bound the constrained optimization problem? If a constraint satisfaction problem $x \in Y$ is NP-Complete.
Can I conclude that the optimization problem $^* =\text{argmax}_x (x)$ s.t. $x \in Y$ is also NP-Complete?
No, no optimization problem can be $\mathsf{NP}$-complete since all $\mathsf{NP}$-complete problems are in $\mathsf{NP}$ and $\mathsf{NP}$ only contains decision problems.