How can one solve this mixed-norm minimization problem subject to a quartic constraint for small N and M (ex: M=12 and N=4):
- $\textbf{c} = [c_{1},...,c_{N}]^T$
- $min\biggl (\|\textbf{c}\|_{1} + \beta\|\textbf{c}\|_{2} ^2\biggr) subject\space to\space \sum\limits_ {m=1} ^M (|v_ {m} |^2-\textbf{c}^H\textbf{a}_m\textbf{a}_m^H\textbf{c}) ^2\leq\epsilon$
Thank you for advance for any suggestions.