The following 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
1, while applying two-phase method.
i) We can detect whether an LP is infeasible at the end of Phase
1, 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.