What is the LASSO regression model solution where X is not orthonormal? If you have the LASSO with non orthonormal X problem, what is its solution?
In othe words, how can I obtain the $$\beta$$ weight values?
We need the mathematical derivation in order to front another, more complex, problem.

(None of this relies on (or is helped by) $X$ being orthonormal.)
Pure ridge regression has a closed-form solution.
$$
\hat\beta_{ridge}=(X^TX+\lambda I)^{-1}X^Ty
$$
However, LASSO regression does not.
LASSO is a special case of elastic net, so if we had a closed-form solution for elastic net, we would have a closed-form solution for LASSO. Since there is no closed-form solution for LASSO, there cannot be a closed-form solution for elastic net.
Much like the maximum likelihood estimation of logistic regression parameters, LASSO and elastic net have their parameters estimated numerically to find:
$$
\hat\beta_{elastic} =\underset{\beta}{argmin} \bigg\{
\vert\vert
y-X\beta
\vert\vert^2
+
\lambda_2\vert\vert
\beta
\vert\vert_2^2
+
\lambda_1\vert\vert
\beta
\vert\vert_1^2
\bigg\}
$$
Sorry to disappoint…
