# Optimization of Frobenius Norm and Nuclear Norm

How to solve the following optimization problem, $$\boldsymbol{\hat{x}} = argmin_{\boldsymbol{X}} \frac{1}{2} \| \boldsymbol{X - Y} \|_F^2 + \lambda \| \boldsymbol{X} \|_{*}$$ where $F$ denotes the Frobenius norm and $*$ denotes the nuclear norm. $\boldsymbol{Y}$ and $\lambda$ are known. $\boldsymbol{X},\boldsymbol{Y} \in C^{N \times M}$.

• are you sure that you need to write it $||x-y||^2_F$ instead of $||x-y||^2_2$. I have seen that people use L2-norm instead of Frobenius norm. – user2806363 Jan 5 '17 at 11:15
• Are you sure you need it in the Complex Domain? – Royi Mar 10 '18 at 14:18