I am considering the following regularized binary quadratic optimization problem with a sparsity penalty
$$ \min_{{\bf x} \in \{\pm 1\}^n} \; {\bf x}^\top {\bf C} \, {\bf x} + \| {\bf A} {\bf x} - {\bf b} \|_1 $$
where the matrix ${\bf A} \in {\Bbb R}^{m \times n}$, the vector ${\bf b} \in {\Bbb R}^m$ and the symmetric matrix ${\bf C} \in {\Bbb R}^{n \times n}$ are given.
I have a very simple idea to lift $\bf x$ to the rank-$1$ matrix
$$ {\bf X} = \begin{bmatrix} 1 \\ {\bf x} \end{bmatrix} \begin{bmatrix} 1 & {\bf x}^\top \end{bmatrix},$$
which gives the equivalent objective $ \operatorname{tr} ({\bf C} {\bf X}) + \| {\bf A} \, {\bf X}_{1:n,0} - {\bf b} \|_1$. Dropping the equality constraint $\operatorname{rank}(X) = 1$ yields the convex relaxation
$$ \begin{aligned} \min_{{\bf X}} \quad & \operatorname{tr} ({\bf C} {\bf X}) + \| {\bf A} \, {\bf X}_{1:n,0} - {\bf b} \|_1 \\ \text{s.t.} \quad & X_{ii}=1,\; {\bf X} \succeq 0. \end{aligned} $$
This is, of course, convex.
Question. Is this the right way to handle sparsity? I tried searching for a related method but couldn't find it. Any pointers would be appreciated.
Motivation
The problem actually arises from an estimation scenario, where the loss function has two parts. The trace term represents the observational error, while the $\ell_1$ norm reflects our prior knowledge that some quantity in the real system should be sparse. In my example, this approach does not yield good results because $\bf A$ and $\bf b$ are imperfect. This leads to a situation where, even if our estimate $\hat{{\bf q}}$ equals the true ${\bf q}^*$, the vector ${\bf A} {\bf q} - {\bf b}$ will not exhibit a sparse structure.
Related
