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Consider the following problem: $$ \text{min} \ f(x) \\ \text{subject to} \ h(x) = 0 $$

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 $$ (1) \ \ \text{min} \ f(x) + \frac{1}{2} \rho \sum_{i=1}^{m} (g_{i}(x))^2 \\ (2) \ \ \text{min} \ f(x) + \rho \sum_{i=1}^{m} |g_{i}(x)|$$

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

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