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 $0\leq y\leq 1$.