I have a problem of finding the maximal value of a norm as follows:
\begin{equation}\max\,\|\boldsymbol x\|_1 \end{equation} $$\text{subject to: } A\boldsymbol x\leq b$$
where $\boldsymbol x=(x_1,x_2,...,x_n)^\top\in \mathbb R^n$ and $\|\boldsymbol x\|_1=\sum_{i=1}^n|x_i|$.
Is it correct that I can find the answer by taking the largest result among that of the following problems:
$$\max\,\boldsymbol \alpha \boldsymbol x$$ $$\text{subject to: } A\boldsymbol x\leq b \text{ and }\boldsymbol \alpha \boldsymbol x \geq 0$$
where $\boldsymbol \alpha = (\alpha_1,...,\alpha_n)$ is progessively taken from the finite set $S_\alpha=\{\boldsymbol \alpha\in\mathbb R^n | \;|\alpha_i|=1\}$? In other words, is it equivalent if I break the original problem into $2^n$ LP problems corresponding to the $2^n$ values of $\boldsymbol \alpha\in S_\alpha $?