# Inequality with convex combination

Consider vectors $v_i \in \mathbb{R}^n$, $z_i \in \mathbb{R}^m$, $i = 1,2,\ldots,N$, and matrices $X$ (positive definite), $F$, $G$ (of appropriate dimensions).

Consider $\alpha_i \in \mathbb{R}_{[0,1]}$, such that $\sum_{i=1}^N \alpha_i = 1$ (see the as the coefficients of a convex combination).

Can the inequality

$$\sum_{i, j \neq i}^N \alpha_i \alpha_j v_i^\top X \left( F v_j + G z_j \right) \leq 0$$

be made independent from the $\alpha_i$s?

Note that this is possible for $N=2$ because $\alpha_1 \alpha_2 = \alpha_1(1-\alpha_1)$ and it can be simplified.

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At least: is the condition $$\sum_{i, j \neq i}^N v_i^\top X \left( F v_j + G z_j \right) \leq 0$$ sufficient? –  Adam Apr 30 '12 at 7:42
We may want to think about the problem as $\max_{\alpha_1,...,\alpha_N} (...) \leq 0$. Can we get something interesting by taking the partial derivatives $\partial/\partial \alpha_i$ (and imposing them to be $0$)? –  Adam Apr 30 '12 at 11:56