Farkas lemma

Given an $m\\times n$ matrix $A$ and an $1\\times n$ real row vector $c$ , both with real coefficients, one and only one of the following systems has a solution:

1.
$Ax\\leq 0$ and $cx>0$ for some $n$ -column vector $x$ ;
2.
$wA=c$ and $w\\geq 0$ for some $m$ -row vector $w$ .

Equivalently, one and only one of the following has a solution:

1.
$Ax\\leq 0$ , $x\\leq 0$ and $cx>0$ for some $n$ -column vector $x$ ;
2.
$wA\\leq c$ and $w\\geq 0$ for some $m$ -row vector $w$ .

Remark. Here, $Ax\\geq 0$ means that every of $A x$ is nonnegative, and similarly with the other expressions.

单词	FarkasLemma
释义	Farkas lemma Given an $m\\times n$ matrix $A$ and an $1\\times n$ real row vector $c$ , both with real coefficients, one and only one of the following systems has a solution: 1. $Ax\\leq 0$ and $cx>0$ for some $n$ -column vector $x$ ; 2. $wA=c$ and $w\\geq 0$ for some $m$ -row vector $w$ . Equivalently, one and only one of the following has a solution: 1. $Ax\\leq 0$ , $x\\leq 0$ and $cx>0$ for some $n$ -column vector $x$ ; 2. $wA\\leq c$ and $w\\geq 0$ for some $m$ -row vector $w$ . Remark. Here, $Ax\\geq 0$ means that every of $A x$ is nonnegative, and similarly with the other expressions.
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