I have the following optimization problem in $\mathbf{x} \in \mathbb{R}^{K \times 1}$
$$\min_{\mathbf{x}>0} \quad \|\mathbf{A}\mathbf{x}\|_0 + \alpha \|\mathbf{B}\mathbf{x}-\mathbf{c}\|_2^2$$
where $\mathbf{A}, \mathbf{B} \in \mathbb{R}^{L \times K}$ are (very) fat matrices ($L<K$) and $\mathbf{c} \in \mathbb{R}^{L\times 1}$. Moreover,
$$\mathbf{A} = \left( \begin{array} {cccccccccccccccccc} \mathbf{1}_1 & \mathbf{0} & \cdots & \mathbf{0}\\ \mathbf{0} & \mathbf{1}_2 & \cdots & \mathbf{0} \\ \vdots &\vdots & \ddots&\vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{1}_K \\ \end{array} \right)$$
where $\mathbf{1}_k=(1, 1, \dots, 1)$ is a $1 \times m_k$ vector whose elements are $1$.
