Interior point methods (also referred to as
barrier methods) are a certain class of
algorithms that solves linear and nonlinear
convex optimization problems.
John von Neumann suggested an interior point method of linear programming which was neither a polynomial time method nor an efficient method in practice. In fact, it turned out to be slower in practice compared to
simplex method which is not a polynomial time method. In 1984,
Narendra Karmarkar developed a method for
linear programming called
Karmarkar's algorithm which runs in provably polynomial time and is also very efficient in practice. It enabled solutions of linear programming problems which were beyond the capabilities of the simplex method. Contrary to the simplex method, it reaches a best solution by traversing the interior of the
feasible region. The method can be generalized to convex programming based on a
self-concordant barrier function used to encode the
convex set.