How is this optimal value derived?

Question

$\min (c^Tx) \text{ subject to } a^Tx \le b$ if $c = \lambda a$ for some $\lambda \lt 0$.

Then the optimal value is $c^Tab = \lambda b$

Can someone explain how the optimal value is derived here?

Answer 1

Your objective function becomes $c^\top x = \lambda a^\top x = \lambda (a^\top x)$. From your constraint, you know that

$$ a^\top x \leq b \overset{\lambda < 0}{\Rightarrow} \underbrace{\lambda a^\top x}_{= c^\top x} \geq \lambda b $$

so this means that $\inf_{x : a^\top x \leq b} c^\top x = \lambda b$. Since your program is a minimization, this must be its optimal value.