I'm little bit confused about the data matrices associated with an SDP program. Consider the following SDP program:
I know that the Matrix $X$ must be positive semidefinite i.e. $X \succeq 0$ as the last constraint implies. My question is about the data matrices $C$ and $A_i \forall i \in \{1, \cdots, m\}$. Do they have to be positive semidefinite ? In other words, can we have a matrix $C$ for example that is either negative definite or indefinite ?
We do not need $C$ to be positive semidefinite. It is an extension of linear programming.
$C\cdot X=\sum_{i=1}^n \sum_{j=1}^n C_{ij}X_{ij}$
Note that due to $X$ is symmetric we have $$C_{ij}X_{ij}+C_{ji}X_{ji}=(C_{ij}+C_{ji})X_{ij},$$
we can obtain the same expression using symmetric matrix by using $\frac12(C+C^T)$ instead. Hence, usually for simplicity, we just states that we want $C$ and $A_i$ to be symmetric.