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?

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