I would like to know if some variables in design matrix are correlated then LASSO is convex or not. If you give me a proof for convexity of LASSO and ADAPTIVE lasso, I will be thankful. LASSO is defined by minimizing penalized likelihood function. Then in the context of regression with Gaussian errors we have $$ L_n=||Y-X'\beta||_2^2 +\lambda_n||B||_1 ,\quad \lambda_n \geq 0 $$ So my question is, if some columns of $X$ are correlated, then how can I prove that $L_n$ is convex(if it is convex). It is obvious that my goal is minimizing $L_n$ with respect to $\beta$. Thank you.
If $XX'$ is positive semi-definite, then $L_n$ is convex.
Expand $\left \|Y-X'\beta \right \|_2^2$, we get $F(\beta)=Y'Y-2Y'X'\beta+\beta'XX'\beta$. $F(\beta)$ is convex with respect to $\beta$ when $XX'$ is positive semi-definite. As you know $\| \beta \|_1$ is convex, therefore $L_n$ is the sum of two convex function, it's convex.