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How to choose an arbitrary point from a polytope defined by system of linear inequalities, Ax < b ? (Here A is an m-by-n matrix, x is n-by-1 and b is m-by-1.) I just want to find one ARBITRARY point that satisfies Ax < b and NOT satisfies Ax = b.

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Keyword: linear programming. –  Boris Bukh Nov 15 '10 at 15:17
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Interior-point methods for linear programming give that automatically. –  Warren Schudy Nov 15 '10 at 17:04
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Interior-point methods will find the so-called analytic center of the polytope. Or solve the LP "max s such that $Ax + s \le b$" to maximize the slack. –  Warren Schudy Nov 15 '10 at 17:06
    
the choice of word 'sampling' is unfortunate, although actually sampling from a polytope is an interesting question :). –  Suresh Venkat Nov 17 '10 at 5:55

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I agree with Boris -- you want to solve an LP. If you would like to solve it numerically, just pose it like this: $$ \begin{align} & \min 0 \\ s.t. \quad & Ax + e\leq b \end{align} $$ where $e \in \mathbb{R}^m$ is a vector containing $\epsilon$'s, which is some numerical tolerance.

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