I can't find a relevant definition of a quadratic function. More precisely, in an exercise about numerical optimization, it is said that the function we will focus on is defined as :
\begin{align}
q~\colon~&\mathbb{R}^P\to\mathbb{R}\\
&x\mapsto\frac{1}{2}\left\langle Bx\vert x\right\rangle_{\mathbb{R}^P} - \left\langle c\vert x\right\rangle_{\mathbb{R}^P}
\end{align}
Where $B\in\mathfrak{M}_{P}(\mathbb{R})$ et $c\in\mathbb{R}^P$. It is said that this function is a quadratic function and it is in numerical optimization a very important example since problems involving quadratic functions can be "easily" solved. There is no definition in my textbook, so I wondered if it was possible to have a little explanation on what exactly is a quadratic function and why this particular function $q$ is a quadratic function ?
Thanks you in advance for any information on this topic.
Edit 1 while writing this : I think there is also a semantic issue there, since I read this : https://en.wikipedia.org/wiki/Quadratic_form and I'm note sure if this is related to what I'm dealing with in this exercise
Edit 2 : This exercise is in french and the name used is fonctionelle quadratique, so I translated it to english as quadratic function. I found no relevant definition using both terms in Google.