Consider the following problem: $$ \text{minimize} \ f(x) \\ \text{subject to} \ g(x) = 0 $$
This is a constrained optimization problem. We want to convert it into an unconstrained optimization problem so that we can use Newton's Method, quasi-Newton methods, etc..
So we have $$ (1) \ \ \text{minimize} \ f(x) + \frac{1}{2} \rho \sum_{i=1}^{m} (g_{i}(x))^2 \\ (2) \ \ \text{minimize} \ f(x) + \rho \sum_{i=1}^{m} |g_{i}(x)|$$
Why is (2) exact and (1) not exact?