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.

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.

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.

Mehrotra's predictor-corrector algorithm is a common implementation of an interior point method.

[edit] References

• Nocedal, Jorge, and Stephen Wright (1999). Numerical Optimization. New York, NY: Springer. ISBN 0-387-98793-2.