Interior point method

Interior-point methods (also referred to as barrier methods or IPMs) are a certain class of algorithms that solve linear and nonlinear convex optimization problems. Interior-point methods (also referred to as barrier methods or IPMs) are a certain class of algorithms that solve linear and nonlinear convex optimization problems. John von Neumann suggested an interior-point method of linear programming, which was neither a polynomial-time method nor an efficient method in practice. In fact, it turned out to be slower than the commonly used simplex method. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, which runs in provably polynomial time and is also very efficient in practice. It enabled solutions of linear programming problems that were beyond the capabilities of the simplex method. Contrary to the simplex method, it reaches a best solution by traversing the interior of the feasible region. The method can be generalized to convex programming based on 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 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. 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. The primal-dual method's idea is easy to demonstrate for constrained nonlinear optimization.For simplicity, consider the all-inequality version of a nonlinear optimization problem: The logarithmic barrier function associated with (1) is Here μ {displaystyle mu } is a small positive scalar, sometimes called the 'barrier parameter'. As μ {displaystyle mu } converges to zero the minimum of B ( x , μ ) {displaystyle B(x,mu )} should converge to a solution of (1). The barrier function gradient is