2

With regards to linear optimization using the simplex method, can someone provide an example:

  1. where both the primal and dual of problem A are infeasible
  2. where both the primal and dual of problem B are unbounded

what is the nature of these problems that satisfy these 2 situations? I suppose we should establish that unbounded never means infeasible and vice versa. So therefore problem A and problem B should be different.

Alexander Mills
  • 342

1 Answers1

1

If a dual is unbounded, then its primal is infeasible. Likewise, if a primal is unbounded, then its dual is infeasible. This fundamentally boils down to the weak duality theorem which has been answered here with a similar question. Note, the proof they have there is actually pretty good. Here are some more helpful links to explore weak and strong duality and their differences:

Here’s another example where the the both questions you proposed has been answered. Take note that the latter answer gives a model that you can play around with yourself to test to see if its true

After asking my professor, and looking into it more, I finally found something that may make this problem make more sense and would answer your question:

Consider the Simplex Tableau: \begin{array} {|c|c|} \hline BV & z & \text{Variables} & RHS & RT \\ \hline z & 1 & C_{BV}^TB^{-1}N-C_{NBV}^T & C_{BV}^TB^{-1}b & - \\ \hline x_n & 0 & & & \\ \text{or} & 0 & B^{-1}A & & \\ s_n & \vdots & \text{or} & B^{-1}b & \frac{B^{-1}b}{B^{-1}A_j} \\ \text{or} & 0 & B^{-1}A_j & & \\ e_n & 0 & & & \\ \hline \end{array}

The solution of the Dual is the objective function of this Tableau right under "Variables", and the solution of the Primal is the right-hand-side $B^{-1}b$.

In order for the Primal to be unbounded, and the Dual be infeasible, we need to have an available pivot in the objective function row, but the entire $A_j$ column of that pivot be either negative or zero. If we translate that to the dual at this very same phase in the pivot, it will be infeasible as some constraint condition will be violated (more will be explained on this shortly). Likewise, the opposite of the above is true if we swap the words of primal with dual and dual with primal. This is all contingent on weak duality. Strong Duality is where the Dual and Primal have the same optical objective function output where the Duality Gap is equal to zero.

In each iteration of the Simplex Method it assumes the following:

At any point in the Simplex Method, excluding the basis where the Optimal Solution is obtained, the Dual is infeasible. The Simplex Method is actually striving to make the Dual become feasible where it meets the optimal objective function value for the Primal, as in every pivot before the optimal basis a constraint condition is violated within the Dual.

The Dual Simplex Method assumes the opposite:

and it works to achieve Primal feasibility, under the same grounds where it keeps trying to improve the Primal up to the point it meets the objective function value of the Dual and out of infeasability.