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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 ?

$$\begin{array}{ll} \text{maximize} & \displaystyle\sum_{i < j} w_{ij}x_{ij} + \displaystyle\sum_k w_k \left( \sum_{(u,v)\in C_k} x_{uv} \right)^2\\ \text{subject to} & x_{ij} \in \{0,1\}\\ & X := \{ x_{ij} \} \succeq 0\end{array}$$

Rodrigo de Azevedo
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Arthur
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1 Answers1

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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.

Tom
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