To minimize $x^TAx$ where $A$ is not necessarily positive semi-definite. Let $A\in \mathbb{n\times n}$ be a symmetric matrix. Let $x\in \mathbb{R}^{n\times 1}$ be an unknown vector. 
The problem is 
$$\min \limits_x x^TAx.$$
Since $A$ is an input, I am not sure 
1  it is positive semidefinite (the objective is convex);
2  or it is negative semidefinite (the objective is concave); 
3  or indefinite (the objective is neither concave nor convex. )
Case 1 is simple.
In Case 2, the minimum is infinite. So people may say it is not well defined.
Could any one tell me how should I handle case 3? Is its minimum infinite?
If I change the problem to,
$$\min \limits_x x^TAx$$
where $x\in C$, $C$ is a convex set. 
For example $\sum\limits_i^n x_i=1$; or $|x_i|\le 1$. Is there any way to find $\arg\min\limits_x x^TAx$?

Suppose that there is some $v$ such that $v^TAv < 0$, i.e. $A$ is not positive semidefinite. Then for every $\lambda > 0$, we have $$(\lambda v)^TA(\lambda v) = \lambda^2 (v^T Av) \ \overset{\lambda \to \infty}{\longrightarrow}\ -\infty$$
It follows that if $A$ is not positive semidefinite the problem is unbounded from below. This is in particular true for indefinite matrices.
