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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 $ s.t. \quad x_{ij}=\{0,1\} , \; X=\{ x_{ij}\} \; \text{positive semi-definite} $

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    Usually we relax the integer constraint to real number between 0 and 1 ?2012-10-16

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Seems to me the author is asking for a semidefinite programing solution. The first term of his objective function is like those in standard SDP problems, but he has a second term.