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I have the following constraints:

$$\sum_{1\leq i,j\leq n,\ i\neq j} x_ix_j\geq 0.25$$

$0\leq x_i \leq 1$ for $i=1, \ldots, n$

Is this set convex?

I think so, but $0.25-\sum_{1\leq i,j\leq n} x_ix_j$ is a convex function? or not? Note that I have $x_i$ are all nonnegative.

Can someone give me a reference on these questions. Standard textbook often talks about a function $R^n \rightarrow R$.

Many thanks.


Updated: following some feedbacks after the first post, I realized that I forgot to put $i\neq j$. I meant constraints like $xy\geq 0.2$ and $0\leq x \leq 1$ and $0\leq y\leq 1$.

  • 0
    Isn't your constraint equal to $\big(\sum_{1\le i\le n} x_i\big)^2 \ge 0.25$?2012-04-16
  • 0
    Following @RahulNarain's comment, in the context of the other constraints on $x_i$, the first constraint is equivalent to $(x_1+...+x_n) \geq 0.5$, so yes the set is convex.2012-04-16

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