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I'm little bit confused about the data matrices associated with an SDP program. Consider the following SDP program:

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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 ?

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    $\begingroup$ Yes, positive semidefiniteness is only enforced on $X$, not $C$ $\endgroup$
    – user1936752
    Jul 19, 2021 at 4:20
  • $\begingroup$ @user1936752 thank you for your answer. Do they have to be symmetrical (matrices $C$ and $A$)? $\endgroup$
    – Ibrahim Amer
    Jul 19, 2021 at 4:53
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    $\begingroup$ Due to the symmetry of $X$ one can always make $C$ and $A$ symmetric. $\endgroup$
    – Rammus
    Jul 19, 2021 at 8:06

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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.

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