Are there any results discussing about the relations between the two following optimization problems?
\begin{align} \text{ (P1) } \hspace{0.5cm} \min \hspace{0.5cm} & \langle a, x \rangle\\ \text{ s.t. } \hspace{0.5cm} & ||x||_1=b\\ & x\geq 0 \end{align} and \begin{align} \text{ (P2) } \hspace{0.5cm} \min \hspace{0.5cm} & ||x||_1 \\ \text{ s.t. } \hspace{0.5cm} & \langle c, x \rangle =d\\ & x\geq 0 \end{align} where $a,c,x\in \mathbb R^n$ and $b,d\in \mathbb R$.
Ps. Any related information is also appreciated!