Consider a linear programming model in the usual form ready for applying the simplex method. I understand that having the constraint equations' coefficient matrix $A$ be of full row rank means not having any redundancy in the constraints.

However, will $A$ not being of full row rank cause any problems in the application of the simplex algorithm? For example, if there is a degenerate basic feasible point that has not been preprocessed away, then the simplex algorithm (if unmodified) may go into an infinite loop and never terminate. How about when $A$ is not of full row rank?


Consider the following polyhedron:

$P = \{x\in\mathbb{R}^2| Ax = b, x\ge0 \}$, where $A = \left[\matrix{1&1 \\ 2&2}\right]$ and $b = \left[\matrix{1 \\ 2} \right]$.

Let the function we want to minimize be $c^Tx$, where $c = \left[\matrix{1 \\ 0}\right]$.

It is clear that $x = \left[\matrix{0 \\ 1}\right]$ is the only optimal solution to the problem. However this $x$ is not a basic feasible solution and, since the simplex method iterates over basic feasible solutions, the simplex method wouldn't succeed in finding the answer to the problem.

Further observations: Actually, the simplex method does not make sense if the rank of A does not equal the number of constrictions ($m$), because the method relies heavily on that hypothesis. Also note that the definition of basic feasible solution (http://www.math.chalmers.se/Math/Grundutb/CTH/tma947/1617/lectures/lecture9.pdf, page 4/30) is given in a standard polyhedron with rank $A = m$. Otherwise (if rank $A < m$), the definition is useless, for there is no subset of $m$ linearly independent columns of A, and therefore there is no basic feasible solution.


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