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 $$ \text{minimize } f(x) + \lambda(b-Ax)?$$
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 $$ \text{minimize } f(x) + \lambda(b-Ax)?$$