I'm having difficulties on answering the following questions (first time I'm trying to prove something), any help would be awesome! Thanks in advance.

Q: It is possible to combine the two phases of the two-phase method into a single procedure by the big-M method. Given the linear program in standard form

minimize $c^Tx$ subject to $Ax=b$, $x>0$,

one forms the approximating problem

minimize $c^Tx+M\sum_{i=1}^{m}y_i$ subject to $Ax+y=b$, $x>0$, $y>0$.

In this problem $y=\left ( y_1,y_2,...,y_m \right )$ is a vector of artificial variables and $M$ is a large constant. The term $M\sum_{i=1}^{m}y_i$ serves as a penalty term for nonzero $y_i$’s.

If this problem is solved by the simplex method, show the following:

a) If an optimal solution is found with $y = 0$, then the corresponding $x$ is an optimal basic feasible solution to the original problem.

b) If for every $M > 0$ an optimal solution is found with $y \neq 0$, then the original problem is infeasible.

c) If for every $M > 0$ the approximating problem is unbounded, then the original problem is either unbounded or infeasible.

d) Suppose now that the original problem has a finite optimal value V(∞). Let V(M) be the optimal value of the approximating problem. Show that $V(M) \leqslant V(∞)$.

e) Show that for $M1 \leqslant M2$ we have $V(M1) \leqslant V(M2)$.

f) Show that there is a value $M_0$ such that for $M > M_0$, V(M) = V(∞), and hence conclude that the big−M method will produce the right solution for large enough values of $M$.


closed as off-topic by Edward Jiang, user296602, colormegone, user147263, user223391 May 8 '16 at 23:44

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I didn't know it, I like this method.

a) If an optimal solution with $y=0$ is found, it is obviously a feasible solution of the original problem.

Let us assume the corresponding $x$ is not optimal regarding the original problem. Then, there would be a feasible $x_0$ such that $c^Tx_0<c^Tx$. Then, $(x_0,y)$ would be a feasible cheaper solution for the new problem. Which is absurd.

Hence $x$ is optimal regarding the original problem.

e) You are looking at a problem with the same constraints, but a cost function always more expensive, so $V(M_1)\leq V(M_2)$

d) The fact that e) is true and the original problem is bounded implies $V(n), n \in \mathbb{N}$ is a monotonous bounded series, so converges.


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