Consider the standard form polyhedron P = {Z ER" Ac = b, < >0}, where A is mxn with linearly independent rows. For each of the following statements, state whether it is true or false. If true, provide a formal proof. If false, provide a clear counterexample.
(a) If n = m +1, then P has at most two basic feasible solutions.
(b) The set of all optimal solutions is bounded.
(c) At every optimal solution, no more than m variables can be positive.
(d) If there is more than one optimal solution, then there are infinitely many optimal solutions.
(e) If there are several optimal solutions, then there exist at least two basic feasible solutions which are optimal.
(f) Consider the problem of minimizing f () = max{cr, d'<} over the set P. If this problem has an optimal solution, then it must have an optimal solution which is an extreme point of P.