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5.23 Strong duality in linear programming. We prove that strong duality holds for the LP minimize $c^T x$ subject to $Ax \ge b$ and its dual maximize $-b^T z$ subject to $A^T z + c = 0$, $z \ge 0$, provided at least one of the problems is feasible. In other words, the only possible exception to strong duality occurs when $p^* = \infty$ and $d^* = -\infty$. (c) Consider the example minimize $x$ subject to $\begin{bmatrix} 0\\1 \end{bmatrix} x \ge \begin{bmatrix} -1\\1 \end{bmatrix}$. Formulate the dual LP, and solve the primal and dual problems. Show that $p^* = \infty$ and $d^* = -\infty$. solve it using the methods: Lagrangian and dual function (not the knowledge of LP duality)

          5.23 Strong duality in linear programming. We prove that strong duality holds for the LP
minimize $c^T x$
subject to $Ax \ge b$
and its dual
maximize $-b^T z$
subject to $A^T z + c = 0$, $z \ge 0$,
provided at least one of the problems is feasible. In other words, the only possible exception to strong duality occurs when $p^* = \infty$ and $d^* = -\infty$.
(c) Consider the example
minimize $x$
subject to $\begin{bmatrix} 0\\1 \end{bmatrix} x \ge \begin{bmatrix} -1\\1 \end{bmatrix}$.
Formulate the dual LP, and solve the primal and dual problems. Show that $p^* = \infty$ and $d^* = -\infty$.
solve it using the methods: Lagrangian and dual
function (not the knowledge of LP duality)

5.23 Strong duality in linear programming. We prove that strong duality holds for the LP
minimize c^T x
subject to Ax ≥ b
and its dual
maximize -b^T z
subject to A^T z + c = 0, z ≥ 0,
provided at least one of the problems is feasible. In other words, the only possible exception to strong duality occurs when p^* = ∞ and d^* = -∞.
(c) Consider the example
minimize x
subject to < b m a t r i x >
x ≥
< b m a t r i x >.
Formulate the dual LP, and solve the primal and dual problems. Show that p^* = ∞ and d^* = -∞.
solve it using the methods: Lagrangian and dual
function (not the knowledge of LP duality)

Added by Donna B.

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