# Equivalency of lasso problems

In the literature, I've seen the lasso problem phrased as the minimization of: $$\frac12x^tAx-x^tb+\lambda||x||_1$$ or of: $$\frac12||Ax-b||_2^2+\lambda'||x||_1=\frac12x^tA^tAx-x^tA^tb+b^tb+\lambda'||x||_1$$

If $$\lambda=\lambda'=0$$, then the second problem minimizes the norm of the gradient of the first, and the two problems are equivalent when $$A$$ is symmetric positive definite ($$x=A^{-1}b$$ is the solution to both). If $$A$$ is not symmetric positive definite then the first problem is no longer convex, while the second always is and can be treated by the standard coordinate/gradient/conjugate gradient descent methods (finding the solution of minimum norm $$||x||_2$$ when $$A$$ is singular).

How does this work when $$\lambda,\lambda'$$ are nonzero? Are the two problems equivalent for symmetric $$A$$ positive definite under some function $$\lambda'=f(\lambda)$$ (or can we get approximate equivalency of solutions)? It'd be quite convenient to substitute the first problem with the second when strict lasso penalization is wanted (no ridge) and $$A$$ is very ill-conditioned.