Let $A$ and $C\ $ be $n\times n$ symmetric matrices, and $A\bullet C=Tr(A^TC)$. Let $S\subseteq [n]\times [n]\times [n]$. Define a non-convex programming problem as follows.
\begin{equation} \begin{split} &\max\ \ \ C\bullet X\\ &\text{ s.t.} \ \ \ \ \ A\bullet X=1,\\ &\qquad \quad X_{i,j}X_{i,k}=0, \ \ \forall\ (i,j,k)\in S,\\ &\qquad \quad X\succeq 0,\ X\geq 0, \end{split} \end{equation} where $X$ is a variable ranging over the set of $n\times n\ $ symmetric matrices over $\mathbb{R}$, $\ X\succeq 0$ means $X$ is positive semi-definite, $\ X\geq 0$ means each entry of $X$ is nonnegative, and $X_{i,j}$ is the $(i,j)$-entry of $X$.
Could you please tell me how to solve this problem? And how to consider its dual problem? Thank you.