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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?

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Ok, what exactly is $g_i(x)$ and where did $h(x)$ go? – TenaliRaman May 4 '12 at 20:22

Neither of the unconstrained reformulations are "exact" in general, in the sense that neither the solution to (1) nor the one to (2) will be the same as the one of the original constrained problem. In fact, solutions to (1) and (2) may be infeasible for the original problem.

What one can say is that, for increasing (positive) values of the penalty parameter, the solutions to (1) and (2) will approach the true solution.

What changes is probably the way the solution is approached, and of course the viable numerical ways for solving the two reformulations.

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