In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on both sides. The significant difference between interior point and simplex based methods is reflected not only in the theoretical background but also in the practical implementation. In this paper we give an overview of the most important characteristics of advanced implementations of interior point methods. First, we present the infeasible-primal-dual algorithm which is widely considered the most efficient general purpose IPM. Our discussion includes various algorithmic enhancements of the basic algorithm. The only shortcoming of the "traditional" infeasible-primal-dual algorithm is to detect a possible primal or dual infeasibility of the linear program. We discuss how this problem can be solve...

The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large-- scale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, but the initialization problem remained unsolved. In this paper we show that the initialization strategy of embedding the problem in a self--dual skew--symmetric problem can also be extended to the semi--definite case. This way the initialization problem of semi--definite problems is solved. This method also provides solution for the initialization of quadratic programs and it is applicable to more general convex problems with conic formulation. Key words: Semidefinite programming, complementarity, skew--symmetric embedding, initialization, self--dual problems, central path. iii 1 Introduction The extension of interior point algorithms from linear programming (LP) to semidefinite programmi...

In this paper we propose a method for linear programming with the property that, starting from an initial non-central point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Along with the convergence analysis we provide a general framework which enables us to analyze various primal-dual algorithms in the literature in a short and uniform way.

This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we introduce the notions of weak/strong feasibility or infeasibility for a general primal-dual pair of conic convex programs, and then establish various relations between these notions and the duality properties of the problem. In the second half of the paper, we propose a self-dual embedding with the following properties: Any weakly centered sequence converging to a complementary pair either induces a sequence converging to a certificate of strong infeasibility, or induces a sequence of primaldual pairs for which the amount of constraint violation converges to zero, and the corresponding objective values are in the limit not worse than the optimal objective value(s). In case of strong duality, ...

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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...

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, ane 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 rst author is supported by the Dutch Organization for Scientic Research (NWO), grant 611-304-028. The fourth author is supported by the Swiss National Foundation for Scientic Research, grant 12-34002.92. ii 1...