I am to optimize the following function
$$\forall x \in \mathbb R^{n}$$
define $x^2$ as: $$x^2=diag(x)x$$
example
$
\mathbb {diag}(
\begin{pmatrix}x_1\\ x_2\\ x_3\end{pmatrix})
\begin{pmatrix}x_1\\ x_2\\ x_3\end{pmatrix}=
\begin{pmatrix}x_1&0&0\\ 0&x_2&0\\ 0&0&x_3\end{pmatrix}\begin{pmatrix}x_1\\ x_2\\ x_3\end{pmatrix}
=
\begin{pmatrix}{x_1}^2\\ {x_2}^2\\ {x_3}^2\end{pmatrix}$
Then:
$$\min\ \ {f(x)}:=c^\top x+\frac 12 {{(x^2)}^\top A{x^2}}$$
but $A$ is not positive semidefinite, but it is symetric. i.e
$$A=A^\top$$
and
$$
\exists z \in \mathbb R^n$$
where:
$$ z^\top Az\not\in\{\mathbb R^+,0\}$$
Infact I define $A$ as
$$A=\begin{pmatrix}
0&1&0&0&\cdots\\
1&0&0&0&\cdots\\
0&0&0&1&\cdots\\
0&0&1&0&\cdots\\
\vdots&\vdots&\vdots&\vdots&\ddots
\end{pmatrix}$$
So that the non matrix representation of the problem is $$\min\ \ \ c^\top x + \sum_{i=0}^{n-1}{(x_{i}x_{i+1})}^2$$
Subject to:
$$x^\top C=b$$
where $b \in \mathbb R^m$ and $C \in \mathbb R^{m\times n}$
Clearly the program has a global minimum and cannot be called non convex but rather weak/non-strong convex. Can there be a polynomial time algorithm for this kind of problem (probably through gradient descent methods)? Actually what could be the bottleneck of finding such solutions if they are hard to get? In my opinion Interior point methods could tackle this problem in polynomial time but how?
Note The above questions are not different.
I realise the question is NP Hard(But i am still not sure so you may as well answer according to how you view it). since no one could answer, I will answer it myself. Take $a_i \in \{0,1\} \forall i \in \mathbb N$. Define the following equations. $$a_u+a_v=1\\a_u\cdot a_v=0$$ Let $$f(x,y):=a_x\cdot a_y$$Therefore$$f^2(x,y):=a_x^2\cdot a_y^2$$ It's easy to see $$ (\min f)\varinjlim-\infty$$ while $$(\min f^2) \varinjlim 0$$ but $a_v\cdot a_u=a_v^2\cdot a_u^2$.
$f^2$ has a defined trough while $f$ doesn't. so we replace the second constraint($f$) with $f^2$. So what we've essentially done is defined an objective function that minimizes towards binary field. Since minimization on binary fields is NP Hard therefore the above minimization cannot be done efficiently in polynomial time unless $P=NP$. Which implies $$\min \sum_{i=0}(a_ia_{i+1})^2\\ \sum_{i=0}f^2(i,i+1)$$ according to the expansion Is NP hard.