A linear program with a single constraint minimize
$z = c_{1}x_{1} + c_{2}x_{2} +· · ·+c_{n}x_{n}$
subject to $a_{1}x_{1} + a_{2}x_{2} +· · ·+a_{n}x_{n} ≤ b$, $x_{1}, x_{2}, . . . , x_{n} ≥ 0.$
(a) Under what conditions is the problem feasible?
(b) Develop a simple rule to determine an optimal solution, if one exists
For part (a), I think only when the problem doesn't degenerate or cycle, and the basis matrix B is invertible, then the problem is feasible?
For part (b), since linear programming is always convex, only need to check it at the "corner point" or on the boundary. This occurs when one or some of $x_{i}=0$ and the remaining variables = 0. Is this rule sufficient?
the problem is always feasible if $b \geq 0$, and is feasible for $ b < 0$ if and only if $a_i < 0$ for some $i$. (However, if for any $a_i < 0$ and $c_i < 0$ for any $i$, z is unbounded below and thus no solution is defined.)