(1) \begin{equation}\label{constrained} \begin{array}{cl} \arg \min \limits_{x} & \|Ax-b\|_2\\ \mathrm{s.t.} & \|x\|_1\le \epsilon \end{array} \end{equation}
(2) \begin{equation}\label{unconstrained} \begin{array}{cl} \arg \min \limits_{x} & \|x\|_1+\lambda\|Ax-b\|_2 \end{array} \end{equation}
Where $\lambda$ is the Lagrange multiplier. Are the two optimization problems equal? If so, how to prove this?