The following questions are to be answered TRUE or FALSE. Please indicate your answer clearly and provide
justification for it.
(a) For any given LP, if x∗ is a feasible solution that is not optimal, then there exists an improving, feasible direction at x∗.
(b) For a given LP in standard form and a given basis matrix B, if both the primal and dual solutions corresponding to B are feasible, then they are both optimal.
(c) For a given LP in standard form and a given basis matrix B, if the dual solution corresponding to B is infeasible, then the primal solution cannot be optimal.
(d) For any given LP, if there exists an optimal solution, there exists an optimal solution that is basic feasible.
(e) A self-dual LP (a problem whose dual is itself) can be unbounded.
(f) For an LP in standard form, no more than m variables can be positive at any optimal solution, where m is the number of equality constraints.
(g) An iteration of the simplex method may move the feasible solution by a positive distance while leaving the cost unchanged.
(h) We can detect whether an LP is unbounded at the end of Phase I, while applying two-phase method.
(i) We can detect whether an LP is infeasible at the end of Phase I, while applying two-phase method.
(j) While running the simplex method, the reason that minimum ratios are used to determine the departing variable is to reach the optimal solution as fast as possible.