Formulation and solution of a minimization problem that consists of a binary variable multiplied by a convex function.

Question

I have the following optimization problem which can be solved using convex optimization toolbox after some relaxations:$${\mathcal{P1}}:{\rm{ }}\mathop {\min }\limits_{\left\{ \mathbf{A}_{i} \right\}_{i = 1}^L,\left\{ \mathbf{B}_{i} \right\}_{i = 1}^L} \sum\limits_{i = 1}^L rank(\mathbf{C}_{i})$$ $$s.t\;\;\;rank(\mathbf{D}_{i})=d$$where $L$ is the number of users and $\mathbf{C}_{i}$ and $\mathbf{D}_{i}$ are functions of ${\left\{ \mathbf{A}_{i} \right\}_{i = 1}^L,\left\{ \mathbf{B}_{i} \right\}_{i = 1}^L}$. Now if the number of users is $N$ instead of $L$ and only $L$ users can participate in the system, so I want to select the optimum set of $L$ users out of $N$ that minimizes the objective function. I formulated the new problem as the following by adding a binary variable: $${\mathcal{P2}}:{\rm{ }}\mathop {\min }\limits_{\left\{ \mathbf{A}_{i} \right\}_{i = 1}^L,\left\{ \mathbf{B}_{i} \right\}_{i = 1}^L,\left\{ \mathbf{S}_{i} \right\}_{i = 1}^N} \sum\limits_{i = 1}^N S_irank(\mathbf{C}_{i})$$ $$s.t\;\;\;rank(\mathbf{D}_{i})=d$$ $$\sum\limits_{i = 1}^NS_i=L$$ $$S_i\in\left\{0,1\right\}$$Is the formulation of $\mathcal{P2}$ correct? If so, how can I solve this problem?