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A draft research paper claims that $Q(p)=1-p_1 p_2 p_3 p_4 - p_2 p_3 p_6 p_7-p_1p_2$ is multilinear where $p_i = \mathbb P(e_i)$ and $e_i$ is a basic event of a component to fail.

I have learnt in LP course that even a function $Q_2(p)=p_1 p_2$ is quadratic when $p_1,p_2 \in \{0,1\}$, not LP. I find the part "linear" confusing in the word "multilinear". The claim is so that $Q(p)$ is a multilinear function. What does this "multilinear" mean? Is this quadratic constraint $Q_2(p)$ multilinear?

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Simply, a function with several variables is called multilinear function if it's linear for each variable.

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I agree. So the function given is not multilinear. –  GEdgar Mar 11 '13 at 21:33
    
Suppose $p_i\in[0,1]$. Now is the function $e^{p_1}$ linear where $e$ is the Euler number? What about $p_1 p_2$? Some examples and counter-examples could help... –  hhh Mar 11 '13 at 21:34
    
$Q$ is not multilinear but $Q_2$ it is so we say also $Q_2$ is bilinear. –  Sami Ben Romdhane Mar 11 '13 at 21:36
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Because there's the $1$, so you can check for example that $Q(p_1+p'_1,p_2,p_3,p_4)\neq Q(p_1,p_2,p_3,p_4)+Q(p'_1,p_2,p_3,p_4)$. –  Sami Ben Romdhane Mar 11 '13 at 21:40
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I'm sorry.en.wikipedia.org/wiki/Multilinear_map –  Sami Ben Romdhane Mar 11 '13 at 21:50

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