Relaxed optimization problems

Question

The original problem is \begin{align} \min & f(x) \tag{1}\\ \text{s.t.} & \text{constraint 1} \tag{2}\\ & \text{constraint 2} \tag{3}\\ \end{align} However, it is very hard to deal with constraint 2. Therefore, I just solve the objective function f(x) only with constraint 1, and I get the optimal solution $x^*$. Coincidentally, $x^*$ also satisfies the constraint 2. Does it mean that $x^*$ is also the optimal solution of the original problem?

Answer 1

Does $x^*$ satisfy the constraints? Yes, you said so.

Can any other $x$ that satisfies the constraints give you a better objective value? No, because $x^*$ is optimal for constraint (1).

So ...

Answer 2

Yes. If the minimum of $f(x)$ for constraint1 satisfies constraint 2, you can see that any other value of $f(x)$ less than the value so obtained will not lie within in the first constraint.