I have the question regarding the expression of the dual formulation of a linear program. I think it is very simple but I just dont get (1) below
$ \min \ c^Tx \\ \text{s.t.} \ \ Ax \le b $
$L(x,\lambda) = c^Tx + \lambda(Ax-b)$
$g(\lambda) = \inf_x ((c + A^T\lambda)^Tx - b^T\lambda) \\ = -b^T\lambda \ \ \text{if} \ \ A^T\lambda + c = 0 \\ = -\infty \ \ \ \ \text{otherwise} $
It is said that $ g(\lambda) \le c^Tx$.
(1) Why is $-b^T\lambda \le c^Tx$ ?
This is a standard proof for weak duality.
Multiply the constraints in the primal by $\lambda$ (to $\lambda^T Ax \leq b^T \lambda$) and in the dual by $x$ (to $\lambda^TAx + c^T x = 0$) to get $-c^Tx \leq b^T \lambda$.