Question

# 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 a*re 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.

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