I have an optimization variable denoted as ${\bf A\in\mathbb{C}^{100\times 5}}=[{\bf a}_1\hspace{1mm} {\bf a}_2 \hspace{1mm} {\bf a}_3 \hspace{1mm} {\bf a}_4 \hspace{1mm} {\bf a}_5];$
Here, ${\bf a}_1$ is the 1st column of matrix ${\bf A}$, ${\bf a}_2$ is the 2nd column of matrix ${\bf A}$ and so on.
Let ${\bf b}=[||{\bf a}_1||_2 \hspace{1mm}||{\bf a}_2||_2 \hspace{1mm} ||{\bf a}_3||_2 \hspace{1mm} ||{\bf a}_4||_2 \hspace{1mm} ||{\bf a}_5||_2]$ be the vector of norms of all the column vectors in matrix ${\bf A}$.
In case it is beneficial we may also allow ${\bf b}=[||{\bf a}_1||_1 \hspace{1mm}||{\bf a}_2||_1 \hspace{1mm} ||{\bf a}_3||_1 \hspace{1mm} ||{\bf a}_4||_1 \hspace{1mm} ||{\bf a}_5||_1]$, i.e, L1 norm instead of L2 norm.
$\textbf{I want to make the norms zero for as many columns as possible.}$
My objective function is expressed as
$\min\hspace{3mm} ||{\bf b}||_0$
Here, $||\cdot||_2$ is L2 norm operator and $||\cdot||_0$ is L0 norm operator.
Here, norm L0 is defined as the number of nonzero elements.
We know that L0 norm is non convex. How to deal this objective? I mean linearization/convexification or approximation.
Or any other means to achieve the same.