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  "body": "\n## Definition\n\nLet $A$ be a real $m \\times n$ [[matrix|0VD9ZSWWJG]], let $b \\in \\R^m$ and let $c \\in \\R^n$.\nFor vectors $u, w \\in \\R^m$, write $u \\le w$ when $u_i \\le w_i$ for every $i$.\nThe *linear program* given by $(A, b, c)$ is the problem\n$$\\text{maximize } c^T x \\quad \\text{subject to} \\quad A x \\le b, \\quad x \\in \\R^n.$$\nA point $x$ with $A x \\le b$ is a *feasible solution*, and the set of them is the *feasible region*.\nA feasible solution $x^*$ is *optimal* if $c^T x^* \\ge c^T x$ for every feasible solution $x$, and then $c^T x^*$ is the *optimal value*.\n*Linear programming* is the study and solution of linear programs.\n\n## Remarks\n\nExample: maximize $x + y$ subject to $x \\ge 0$, $y \\ge 0$, $x + 2y \\le 4$ and $3x + y \\le 6$.\nThe feasible region is the quadrilateral with vertices $(0,0)$, $(2,0)$, $(8/5, 6/5)$ and $(0,2)$.\nThe objective takes the values $0$, $2$, $14/5$ and $2$ at these vertices, so the optimal value is $14/5$, attained at $(8/5, 6/5)$.\n\nNon-example: maximizing $x + y$ subject to $x^2 + y^2 \\le 1$ is not a linear program, because the constraint is not a linear inequality.\n\nEvery linear program is of exactly one of three kinds: infeasible, unbounded (the objective is not bounded above on the feasible region), or with an optimal solution.\nOther standard forms, such as maximizing $c^T x$ subject to $A x = b$ and $x \\ge 0$, are equivalent to this one after adding slack variables or splitting free variables.\nThe dual of the program above is to minimize $b^T y$ subject to $A^T y = c$ and $y \\ge 0$.\nIf one of the two programs has an optimal solution, so does the other, with the same optimal value.\n",
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