Comparison between the simplex algorithm and interior point method

The success of primal-dual algorithm and its variants, as well as IPM in general,

comes (or at least was triggered) from its superiority with respect to simplex
algorithm. The primal-dual algorithm has a set of advantages with respect to simplex
algorithm, but also disadvantages that fortunately can be handled within the

algorithm. Some of the advantages and disadvantages are enumerated below.

Advantages of primal-dual algorithm:

• The algorithm does not has any problems with degeneracies and the number of
iterations is not related to the number of vertices in the feasible region.
• For large LP problems the algorithm uses significantly fewer iterations than
simplex algorithm.
• Most implementations of the algorithm usually solve a LP problem in less than

100 iterations even though the problem may contain millions of variables.

Disadvantages of primal-dual algorithm:
• The algorithm cannot detect a possible infeasible or unbounded status of the
problem, and in some sense the primal-dual algorithm is not complete.
Fortunately, this problem can be handled using homogenous model [4,7].
• Each iteration of the primal-dual algorithm is computationally much more
expensive than one iteration of the simplex algorithm. However, the total work
performed to solve a LP problem is a product of the number of iterations and the
work performed in each iteration. For a large LP problem (say more than 100
variables) the primal-dual algorithms outperforms the simplex algorithm, and the
bigger the problem size is this is the more pronounced.