Inequality constraints

Question

How can I prove that,if certain number $x$ minimizes $f(x)$ with constraints $g(x) \leqslant 0 $ ,then it also minimizes the same function with constrains $g(x) = 0$ ?

Answer 1

This is not true.

Let $f(x)=x^2$

Let $g(x)=x-1$ then the global minimizer is $x=0$ which is not feasible for the new feasible region.

Answer 2

If you find a solution $x^*$ to $min f(x) s.t. g(x) \le 0$, and it satisfies $g(x^*) = 0$, then $x^*$ is optimal for $min f(x) s.t. g(x) = 0$.