Interior point method
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Interior point methods (also referred to as barrier methods) are a certain class of algorithms to solve linear and nonlinear convex optimization problems.
These algorithms have been inspired by Karmarkar's algorithm, developed by Narendra Karmarkar in 1984 for linear programming. The basic elements of the method consists of a self-concordant barrier function used to encode the convex set. As opposed to the simplex method, it reaches the optimal vertex 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. The idea of encoding the feasible set using a barrier and designing barrier methods was studied in the early 1960s by Fiacco-McCormick and others. 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).
Nesterov and Nemirovskii 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, in practice 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.
[edit] References
- Wright, Stephen (1997). Primal-Dual Interior-Point Methods. Philadelphia, PA: SIAM. ISBN 0-89871-382-X.
- Nocedal, Jorge; and Stephen Wright (1999). Numerical Optimization. New York, NY: Springer. ISBN 0-387-98793-2.
