Recently, the dual simplex method has attracted considerable interest. This is mostly
due to its important role in solving mixed integer linear programming problems where
dual feasible bases are readily available. Even more than that, it is a realistic alternative
to the primal simplex method in many cases. Real world linear programming problems
include all types of variables and constraints. This necessitates a version of the dual
simplex algorithm that can handle all types of variables efficiently. The paper presents
increasingly more capable dual algorithms that evolve into one which is based on the
piecewise linear nature of the dual objective function. The distinguishing features of
this method are: (i) in one iteration it can make progress equivalent to many traditional
dual iterations, (ii) using proper data structures it can be implemented very efficiently
so that an iteration requires hardly more work than the traditional pivot method, (iii)
its effectiveness just increases if more upper bounded variables are present, (iv) it has
inherently better numerical stability because it can create a large flexibility in finding
a pivot element, (v) it can excel itself in coping with degeneracy as it can bypass dual
degenerate vertices more easily than the traditional pivot procedures. The power of the
method is demonstrated through examples. The performance on some real world problems
is also presented.