(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.

  • $\begingroup$ What is $\mathcal E$ ? $\endgroup$ – dohmatob Apr 7 '16 at 15:51
  • $\begingroup$ Is this really sufficiently different from the other question you posted to warrant a separate post? Sure, the functions are different, but nothing has changed conceptually. $\endgroup$ – Michael Grant Apr 7 '16 at 16:56
  • $\begingroup$ And of course, neither of these problems is convex. $\endgroup$ – Michael Grant Apr 7 '16 at 16:58

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