(1) \begin{equation}\label{constrained} \begin{array}{cl} \arg \min \limits_{\mathcal{C}_k} & \text{rank}(\mathcal{C}_k)\\ \mathrm{s.t.} & \mathcal{E}(\phi_{j}^{k})\le \epsilon \end{array} \end{equation}
(2) \begin{equation}\label{unconstrained} \begin{array}{cl} \arg \min \limits_{\mathcal{C}_k} & \mathcal{E}(\phi_{j}^{k})+\lambda \text{rank}(\mathcal{C}_k) \end{array} \end{equation}
Where $\lambda$ is the Lagrange multiplier. As we know, the Lagrange multiplier method can transform an optimization problem with equality constraints into a unconstrained optimization problem. But, the problem (1) is an optimization problem with inequality constraints.