# Lasso - Representing l1-ball constraint using the penalized formulation [duplicate]

I would like to solve the following problem:

$$\beta = \arg\min_{\beta;\|\beta\|_1 \leq1} \|X\beta-y\|_2^2$$

which happens to be the constrained formulation of the Lasso, considering only parameter vectors that lie inside the $\ell 1$-unit-ball.

However, Matlab uses the penalized formulation, i.e.

$$\beta = \arg \min_{\beta} \|X\beta-y\|_2^2 + \lambda\|\beta\|_1$$

So my question is the following: is there a way to choose $\lambda$, or a way to transform the formulation, so that I can represent the first problem statement (with its constraint) using the second formulation? In other words, can I choose a value of $\lambda$ such that the parameter vector stays within the $\ell 1$-unit-ball?

• For the constraint $t$ in the $\|\beta\|_1 \leq t$ formulation, there is a always an equivalent $\lambda$ formulation. See two related questions. – Affine May 24 '15 at 3:04