1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x

A modification of the (infeasible) primal-dual interior point method is developed. The method uses multiple corrections to improve the centrality of the current iterate. The maximum number of corrections the algorithm is encouraged to make depends on the ratio of the efforts to solve and to factorize the KKT systems. For any LP problem, this ratio is determined right after preprocessing the KKT system and prior to the optimization process. The harder the factorization, the more advantageous the higher-order corrections might prove to be. The computational performance of the method is studied on more difficult Netlib problems as well as on tougher and larger real--life LP models arising from applications. The use of multiple centrality corrections gives on the average a 25% to 40% reduction in the number of iterations compared with the widely used second-order predictor-corrector method. This translates into 20% to 30% savings in CPU time.

Many issues that are crucial for an efficient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primal-dual algorithm is presented. Next, many tricks that make it so efficient in practice are discussed in detail. Those include: the preprocessing techniques, the initialization approaches, the methods of computing search directions (and lying behind them linear algebra techniques), centering strategies and methods of stepsize selection. Several reasons for the manifestations of numerical difficulties like e.g.: the primal degeneracy of optimal solutions or the lack of feasible solutions are explained in a comprehensive way. A motivation for obtaining an optimal basis is given and a practicable algorithm to perform this task is presented. Advantages of different methods to perform postoptimal analysis (applicable to interior point optimal solutions) are discussed. Important questions that still remain open in the implementations of i...

The introduction of Karmarkar's polynomial algorithm for linear programming (LP) in 1984 has influenced wide areas in the field of optimization. While in 80s emphasis was on developing and implementing efficient variants of interior point methods for LP, the nineties have shown applicability to certain structured nonlinear programming and combinatorial problems. We will give a historical account of the developments and outline the major contributions to the field in the last decade. An important class of problems to which interior point methods are applicable is semidefinite optimization, which has recently gained much attention. It has a lot of applications in various fields (like control and system theory, combinatorial optimization, algebra, statistics, structural design) and can be efficiently solved with interior point methods.

In this paper we propose a long--step target--following methodology for linear programming. This is a general framework, that enables us to analyze various long--step primal--dual algorithms in the literature in a short and uniform way. Among these are long--step central and weighted path--following methods and algorithms to compute a central point or a weighted center. Moreover, we use it to analyze a method with the property that it, starting from an initial non-- central point, generates iterates that simultaneously get closer to optimality and closer to centrality. Key words: interior-point method, affine scaling method, primal--dual method, long--step method. Running title: Target--Following Methods for LP. This work is completed with the support of a research grant from SHELL. The first author is supported by the Dutch Organization for Scientific Research (NWO), grant 611-304-028. The fourth author is supported by the Swiss National Foundation for Scientific Research, grant 12-...

Introduction HOPDM is an implementation of the primal--dual interior point method for solving large scale linear programming (LP) problems. HOPDM stands for Higher Order Primal Dual Method. Its newest version 2.12 (of May 1995) is a robust and efficient code that compares favourably with the up to date commercial LP solvers. The development of HOPDM started from our joint work with Anna Altman in 1992 [2, 3] (version 1.0 of the code was made public domain through the ORSEP). Since then the code has been independently developed in two different directions. Motivated with a real life application [4], Altman extended it to handle separable convex quadratic programs [1] while Gondzio [9, 10] concentrated on improving robustness and efficiency of the basic LP algorithm. Fundamentals The algorithm implemented in HOPDM is a new variant of a primal--dual method. The basic primal--dual algorithm has been developed independently by Megiddo [15] and Kojima, Mizuno and Y