I need to minimize a function $v(h)$ of the form
\begin{align} v(h) = \langle g,h \rangle + \langle Ah, h \rangle + a \cdot \|h\|_4^4, \end{align}
where $a \in \mathbb{R}$ is scalar, $g,h \in \mathbb{R}^d$ are vectors, and $A \in \mathbb{R}^{d \times d}$ is a quadratic and symmetric matrix, but not necessarily positive semidefinite, $ \|h\|_4^4 = \sum_{i=1}^n h_i^4 $ is the sum of entries taken to the fourth power element-wise.
The problem is the L4-norm in the last term.
If I want to write down the first derivative in vectorial form, I need to use the Hadamard product to get \begin{align} \nabla v(h) = g + 2Ah + 4a \cdot h \odot h \odot h, \end{align}
but I don't see a way to make use of this expression.
Any help is appreciated.