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In linear programming where we seek to minimize $c^Tx \to \text{min}_{x\in P}!$ with respect to some inequality constraints, why do we choose $P$ in the form

$Ax \leq b$, $x \geq 0$

as the standard form of a linear program? In the geometrical view $\{x: Ax \leq b\}$ seems much more natural, why don't we use this, or a type II-inequality constraint? (I am aware that we can transform the inequalities, but this does not always preserve the dimension of the problem, and gives an overhead to the proofs)

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Exactly who "chooses $P$ in the form $Ax\le b,x\ge 0$"? I for one always use the standard form $\min c^T x: Ax=b, x\ge 0$ as this is the formulation on which I can show the procession of the simplex method to my students most naturally. By the way, I think what you refer to is the canonical form. –  Gabor Retvari Aug 10 '12 at 19:04
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