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Suppose we have the following problem: $ \text{minimize } \ f(x) \\ \text{subject to } \ Ax = b$

How do we know whether to write the Lagrangian Dual as $ \text{minimize } f(x) + \lambda(Ax-b)$ versus $ \text{minimize } f(x) + \lambda(b-Ax)?$

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We don't, and it does not matter. The $\lambda$ you are going to find will change sign. See also here.

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    When you minimize, $(x, \lambda)$ is variable. In one case you'll get $(x_0,\lambda_0)$ as a solution, in the other case $(x_0,-\lambda_0)$. You are interested in $x_0$. The solution is the same, yes. (And both problems are equally difficult).2012-06-15