# LinearProgramming behavior

Consider the following linear programming problem:

LinearProgramming[-{10, -57, -9, -24}, -( {
{0.5, -5.5, 2.5, 9},
{0.5, -1.5, -.5, 1},
{1, 0, 0, 0}
} ), -{0, 0, 1}]
(* {0., 0., 0., 0.} *)


Why is this answer being returned? Taking $\textbf{x}=\begin{pmatrix}1&0&1&0\end{pmatrix}^T$ satisfies $\textbf{x}\ge 0\land A\textbf{x}\ge \textbf{b}$, but the value of the objective function for this solution is $-1$, which is clearly smaller than $0$, the suggested "optimal" solution by Mathematica.

I would expect Mathematica to be able to solve such a small LP problem, even if it is degenerate. Am I doing something wrong here?

You need to look at what LinearProgramming does: it finds a vector x that minimizes the quantity c.x subject to the constraints m.x>=b and x>=0. For your candidate answer x={1,0,1,0}, the product m.x is equal to {-3., 0., -1.}, which is not greater than your b={0, 0, -1}. Hence this is not a valid solution. Indeed, x={0,0,0,0} is the best you can get given the constraints.