The Lagrangian form of basis pursuit denoising
$\min_{w} ||w||_{\ell_{1}} + \lambda ||Aw-x||_{\ell_{2}}^{2}$
can be solved using proximal gradient descent. Proximal methods also can be used to solve
$\min_{w} ||w||_{\ell_{1}} \text{ s.t. } ||Aw-x||_{\ell_{2}}^{2} \le \epsilon$
by solving the Lagrangian form and iterating over $\lambda$ until the constraint (complementary slackness) is met. But this could result in having to solve the Lagrangian form multiple times (for different $\lambda$) before finding the correct $\lambda$? Is there a better approach to solving this problem?