optimal solution for different feasibility sets

Question

Suppose we have an optimal solution $x^*$ for the convex optimization problem $ \min_x \{f(x) : Cx \leq d\}$ and we know that $x^* \in \{x : Ax=b, Cx\leq d\}$. The function $f(x)$ is a strictly convex quadratic function.

Can we say that $x^*$ is also the optimal solution for $\min_x \{f(x) :Ax=b, Cx \leq d\}$?

Answer 1

I think we can, since $Ax=b$ will add nothing but some extra constraints which are already satisfied bu $x^*$.