Let $X=\{x_{ij}\} \in R^{n \times n}$ denote a variable matrix, and $C_k,k=1,\ldots,m$ denote subsets of $\{(i,j):i=1,\ldots,n, \quad j=1,\ldots,n\}$, while $w_{ij}$ and $w_k$ are constants. The following optimization problem seems not a standard semi-definite programming since it is maximizing a convex function rather than minimizing it. So how to solve it ?
$max \quad \sum_{i,j, i