Let $A$ be an $n \times n$ strictly positive definite matrix with strictly positive entries. Let $c \in \mathbb{R}^n$ be an arbitrary vector.
$$\begin{array}{ll} \underset{x \in \mathbb{R}^n}{\text{minimize}} & (x-c)'A(x-c) =: f (x)\\ \text{subject to} & x \in [0,\infty)^n \end{array}$$
where the symbol $'$ stands for transposition. Assume that $n$ is a large integer. It is true that the solution to the above problem, say $x^*$, is such that $x_{i}^* c_i\geq 0$ for all $i\in [n]$?
Here, $x_{i}^*$ and $c_i$ denote the $i^{th}$ element of $x^*$ and $c$.
This is tantamount to asking whether $x_{i}^*=0$ when $c_i\leq 0$ and $x_{i}^*\geq 0$ when $c_i\geq 0$. If this is not the case, can we find additional conditions on $A$ so that this is true?