If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?
Multiple solutions for both primal and dual
2 Answers
In page 147 of (Sierksma, 2001), an example (Example 2) is given for this case. That is, "In case of (a), a dual optimal solution can either be multiple or non-multiple ...".
Primal Optimal Solution Dual Optimal Solution (a) Multiple implies Degenerate (b) Unique and nondegenerate implies Unique and nondegenerate (c) Multiple and nondegenerate implies Unique and degenerate (d) Unique and degenerate implies Multiple
Sierksma, (2001) Linear and Integer Programming: Theory and Practice, Second edition.
The example is as follows:
\begin{align} (P)&&&\min{}& &\quad{} 5x_2 \\ &&& \text{s.t.}& & - 2x_2 \geq{} 0 \\ && &&x_1& + 4x_2 \geq{} 3 \\ &&&& x_1&,\phantom{+4} x_2 \geq{} 0 \\ \end{align}
\begin{align} (D)&&&\max{}& &\quad{} 3y_2 \\ &&& \text{s.t.}& &\phantom{4}\quad{} y_2 \leq{} 0 \\ && &&-2y_1& + 4y_2 \leq{} 5 \\ &&&& y_1&,\phantom{+4} y_2 \geq{} 0 \\ \end{align}
The solution for $(P)$ is $(x_1, x_2)=(a, 0)$ with $a\geq{}3$ where the solution of $(D)$ is $(y_1, y_2) = (b, 0)$ with $b\geq{}0$. Therefore, yes, it is possible for both primal and dual to have multiple optimal solutions.
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1https://math.stackexchange.com/questions/1049796/prove-optimal-solution-to-dual-is-not-unique-if-optimal-solution-to-the-primal-i – 2017-11-09
According to the table you have mentioned, there is no record for multiple implies multiple, every multiple solution will be expressed as degenerate in dual. Therefore, this is impossible that both have multiple solution.
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0https://math.stackexchange.com/questions/1049796/prove-optimal-solution-to-dual-is-not-unique-if-optimal-solution-to-the-primal-i – 2017-11-09