Interior Point Method
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. The idea of encoding the feasible set using a barrier and designing barrier methods was studied in the early 1960s by, amongst others, Anthony V. Fiacco and Garth P. McCormick. 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 Arkadii 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.
Read more about Interior Point Method: Primal-dual Interior Point Method For Nonlinear Optimization
Famous quotes containing the words interior, point and/or method:
“Men expect too much, do too little,
Put the contraption before the accomplishment,
Lack skill of the interior mind
To fashion dignity with shapes of air.
Luxury, yes but not elegance!”
—Allen Tate (1899–1979)
“... there is no point in being realistic about here and now, no use at all not any, and so it is not the nineteenth but the twentieth century, there is no realism now, life is not real it is not earnest, it is strange which is an entirely different matter.”
—Gertrude Stein (1874–1946)
“Women are denied masturbation even more severely than men and that’s another method of control—they’re not taught to please themselves.... Most women—it takes them a while to warm up to the “situation” but once they get into it, I’m sure they’re going to get just as hooked as—well, everyone I know is!”
—Lydia Lunch (b. 1959)