Question

All linear programs are either unbounded, infeasible, or have an optimal solution.

Is it possible to have a linear program with constraints Ax ≤ b and x ≥ 0 such that, just by changing the value of b, we can get a linear program of all three types types?

Answer 1

if we have a linear program:

(1) A certificate that a feasible is optimal is the set of dual conditions:

(2) A certificate that this is unbounded is the existence of a feasible x and the determination that implies a contradiction.

(3) A certificate that this is infeasible is the existence of a sequence such that for all and as .

So, you can choose b such that above condition holds to get different types of solution.

NOTE: y is variable for LP's dual.

NOTE: LP with constraints Ax ≤ b and x ≥ 0 can be changed in standard form Ax = b and x ≥ 0. So, make the LP in standard form and then apply above certificates.