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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 ?

$$max \quad \sum_{i,j, i<j} w_{ij}x_{ij} + \sum_k w_k(\sum_{(u,v)\in C_k} x_{uv})^2 $$ $$ s.t. \quad x_{ij}=\{0,1\} , \; X=\{ x_{ij}\} \; \text{positive semi-definite} $$

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Usually we relax the integer constraint to real number between 0 and 1 ? – Arthur Oct 16 '12 at 3:14

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.

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