Interior Point Method Algorithm
An Interior Point Method Algorithm is a convex optimization algorithm that uses a self-concordant barrier function to encode the convex set.
- AKA: Barrier Algorithm.
- Context:
- It can be a Linear Programming Algorithm by solving a Linear System (in [math]\displaystyle{ O(n^3) }[/math]).
- …
- Counter-Example(s):
- See: Nesterov Algorithm, Barrier Function, Convex Optimization, John Von Neumann, Karmarkar's Algorithm, Narendra Karmarkar.
References
2014
- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Interior_point_method Retrieved:2014-9-28.
- Interior point methods (also referred to as barrier methods) are a certain class of algorithms to solve linear and nonlinear convex optimization problems.
The interior point method was invented by John von Neumann. Von Neumann suggested a new method of linear programming, using the homogeneous linear system of Gordan (1873) which was later popularized by Karmarkar's algorithm, developed by Narendra Karmarkar in 1984 for linear programming. The method consists of a self-concordant barrier function used to encode the convex set. Contrary to the simplex method, it reaches an optimal solution by traversing the interior of the feasible region. Any convex optimization problem can be transformed into minimizing (or maximizing) a linear function over a convex set by converting to the epigraph form. The idea of encoding the feasible set using a barrier and designing barrier methods was studied by Anthony V. Fiacco, Garth P. McCormick, and others in the early 1960s. These ideas were mainly developed for general nonlinear programming, but they were later abandoned due to the presence of more competitive methods for this class of problems (e.g. sequential quadratic programming). Yurii Nesterov and Arkadi Nemirovski came up with a special class of such barriers that can be used to encode any convex set. They guarantee that the number of iterations of the algorithm is bounded by a polynomial in the dimension and accuracy of the solution. Karmarkar's breakthrough revitalized the study of interior point methods and barrier problems, showing that it was possible to create an algorithm for linear programming characterized by polynomial complexity and, moreover, that was competitive with the simplex method. Already Khachiyan's ellipsoid method was a polynomial time algorithm; however, it was too slow to be of practical interest. The class of primal-dual path-following interior point methods is considered the most successful.
Mehrotra's predictor-corrector algorithm provides the basis for most implementations of this class of methods.
- Interior point methods (also referred to as barrier methods) are a certain class of algorithms to solve linear and nonlinear convex optimization problems.