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:
“Anyone with a real taste for solitude who indulges that taste encounters the dangers of any other drug-taker. The habit grows. You become an addict.... Absorbed in the visions of solitude, human beings are only interruptions. What voice can equal the voices of solitude? What sights equal the movement of a single day’s tide of light across the floor boards of one room? What drama be as continuously absorbing as the interior one?”
—Jessamyn West (1902–1984)
“The point is that nationalism, even in its latest madhouse frenzies under Mussolini and Hitler, is still, like advertising, an arm of big business. Nations as we know them today, were the invention of business and it is natural that business should still consider itself slightly above patriotism and that the strongest international should still be the international of profits.”
—John Dos Passos (1896–1970)
“Too poor for a bribe, and too proud to importune,
He had not the method of making a fortune.”
—Thomas Gray (1716–1771)