In this paper, we show that the moving directions of the primal-affine scaling method (with logarithmic barrier function), the dual-affine scaling method (with logarithmic barrier function), and the primal-dual interior point method are merely the Newton directions along three different algebraic "paths" that lead to a solution of the Karush-Kuhn-Tucker conditions of a given linear programming ...

Interior point methods proved to be efficient and robust tools for solving large–scale optimization problems. The standard infeasible–start implementations scope very well with wide variety of problem classes, their only serious drawback is that they detect primal or dual infeasibility by divergence and not by convergence. As an alternative, approaches based on skew–symmetric and self–dual refo...

Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, secondorder cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of th...

Linear optimization (LO) is the fundamental problem of mathematical optimization. It admits an enormous number of applications in economics, engineering, science and many other fields. The three most significant classes of algorithms for solving LO problems are: Pivot, Ellipsoid and Interior Point Methods. Because Ellipsoid Methods are not efficient in practice we will concentrate on the comput...

In this paper we propose new primal-dual interior point methods (IPMs) for P∗(κ) linear complementarity problems (LCPs) and analyze the iteration complexity of the algorithm. New search directions and proximity measures are defined based on a class of kernel functions, ψ(t) = t 2−1 2 − R t 1 e q “ 1 ξ −1 ” dξ, q ≥ 1. If a strictly feasible starting point is available and the parameter q = log „...

We discuss interior point methods for large-scale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primal-dual interior point method, and consider the numerical issues involved in calculating the search direction in each iteration, including the use of factorization methods and/or preconditioned conjugat...

This paper analyzes sequences generated by infeasible interior point methods. In convex and nonconvex settings, we prove that moving the primal feasibility at the same rate as complementarity will ensure that the Lagrange multiplier sequence will remain bounded, provided the limit point of the primal sequence has a Lagrange multiplier, without constraint qualification assumptions. We also show ...

Hopfield neural networks and interior point methods are used in an integrated way to solve linear optimization problems. The Hopfield network gives warm start for the primal–dual interior point methods, which can be way ahead in the path to optimality. The approaches were applied to a set of real world linear programming problems. The integrated approaches provide promising results, indicating ...