I am working on a paper about optimization using the Huber's Loss function, which is defined as:
\begin{equation} \psi(x)=\begin{cases} \frac{x^2}{2\gamma},& \text{if } \lvert x\rvert\leq\gamma\\ \lvert x\rvert-\frac{\gamma^2}{2},&\text{if } \lvert x\rvert>\gamma \end{cases}. \end{equation} Now suppose we have $ A\in\mathbb{R}^{M\times N} $ and $d\in\mathbb{R}^{M}$ and define the new operator: \begin{equation} \Psi(u)=\sum_{i=1}^M \psi([Au-d]_i). \end{equation} I have proven that $ \Psi:\mathbb{R}^N\rightarrow\mathbb{R}^M $ is continuously differentiable ($ \mathcal{C}^1$) and convex, but not coercive for the directions in $ \text{ker}(A) $. In some sense this operator $ \Psi $ acts like an M-estimator. How can I be sure that: \begin{equation} \min_{u\in\mathbb{R}^N}\Psi(u) \end{equation} always admits a solution? Is there a general existence property for M-estimators satisfying some conditions? Actually I do not need an explicit solution, I am wondering about the existence (i.e. implicit solutions are OK!). Thanks in advance!
