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...

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.

A practical warm-start procedure is described for the infeasible primal-dual interior-point method employed to solve the restricted master problem within the cutting-plane method. In contrast to the theoretical developments in this field, the approach presented in this paper does not make the unrealistic assumption that the new cuts are shallow. Moreover, it treats systematically the case when a large number of cuts are added at one time. The technique proposed in this paper has been implemented in the context of HOPDM, the state of the art, yet public domain, interior-point code. Numerical results confirm a high degree of efficiency of this approach: regardless of the number of cuts added at one time (can be thousands in the largest examples) and regardless of the depth of the new cuts, reoptimizations are usually done with a few additional iterations. Key words. Warm start, primal-dual algorithm, cutting-plane methods. Supported by the Fonds National de la Recherche Scientifique Su...

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...

We addres some of the issues that arise when an interior point method is used to handle the master problem in a decomposition approach. The main points concern the efficient exploitation of the special structure of the master problem to reduce the cost of a single interior point iteration. The particular structure is the presence of GUB constraints and the natural partitioning of the constraint matrix into blocks built of cuts generated by different subproblems. The method can be used in a fairly general case, i.e., in any decomposition approach whenever the master is solved by an interior point method in which the normal equations are used to compute orthogonal projections. Computational results demonstrate its advantages for one particular decomposition approach: Analytic Center Cutting Plane Method (ACCPM) is applied to solve large scale nonlinear multicommodity network flow problems (up to 5000 arcs and 10000 commodities). Key words. Convex programming, interior point methods, cutt...

In this paper, we generalize the notion of weighted centers to semidefinite programming. Our analysis fits in the v-space framework, which is purely based on the symmetric primal-dual transformation and does not make use of barriers. Existence and scale invariance properties are proven for the weighted centers. Relations with other primal-dual maps are discussed. Key words. semidefinite programming, symmetric primal--dual transformation, weighted center. 1 Econometric Institute, Erasmus University Rotterdam, The Netherlands, sturm@few.eur.nl. 2 Econometric Institute, Erasmus University Rotterdam, The Netherlands, zhang@few.eur.nl Sturm and Zhang: On weighted centers for SDP 1 1. Introduction The central path plays a fundamental role in the interior point methodology, both for linear and semidefinite programming. Megiddo [10] showed some highly interesting properties of the central path for linear programming. The fact that ¯-centers are the minimizers of the logarithmic barrier f...

Several algorithms in optimization can be viewed as following a solution as a parameter or set of parameters is adjusted to a desired value. Examples include homotopy methods in complementarity problems and path-following (infeasible-) interior-point methods. If we have a metric in solution space that corresponds to the complexity of moving from one solution point to another, there is an induced metric in parameter space, which can be used to guide parameteradjustment schemes. We investigate this viewpoint for feasible- and infeasible- interior-point methods for linear programming. Key words: homotopy methods, path-following methods, interior-point algorithms, linear programming, Riemannian metrics Running Header: Adjusting Parameters in Homotopy Methods 1 Introduction This paper is concerned with developing guidelines for optimal adjustment of parameters in homotopy or path-following algorithms in optimization, concentrating on interior-point methods for linear programming. The gen...

) Raphael Hauser Primal-dual interior point methods for convex optimization problems are designed to solve a problem and its dual jointly by making use of convex duality theory. In fact, such algorithms usually aim at reducing the duality gap between primal and dual approximate solutions to zero. Primal-dual methods are most conveniently studied in a framework that exhibits the symmetry between the two optimization problems jointly solved by the algorithm. In the linear programming literature such a framework, known as the V -space approach, allows for a unified running-time analysis of several important classes of primal-dual interior-point algorithms. The unified class of algorithms is known as target-following algorithms (see e.g. [4, 2, 3]). The present article is the second in a series of papers that aim at extending this unified theory to self-scaled conic programming, a class of convex optimization problems which contains linear-, semidefinite- and convex quadratic programming w...

Square-root fields are differentiable operator fields used in the construction of target direction fields for self-scaled conic programming, a unifying framework for primal-dual interior-point methods for linear programming, semidefinite programming and secondorder cone programming. In this article we investigate square-root fields for so-called isotropic self-scaled barrier functionals, i.e. self-scaled barrier functionals that are invariant under rotations of their conic domain of definition. We prove a structure theorem for so-called congruent square-root fields for isotropic self-scaled barrier functionals in terms of the irreducible decomposition of their domain of definition. Using this structure theorem, we then investigate primal-dual symmetry and scale-invariance of such square-root fields. In our main theorem we show that these two assumptions together with one additional natural invariance property (so-called canonical reduction) dramatically reduce the degree of freedom in...

The theory of self-scaled conic programming provides a unified framework for the theories of linear programming, semidefinite programming and convex quadratic programming with convex quadratic constraints. The standard search directions for interiorpoint methods applied to self-scaled conic programming problems are the so-called Nesterov-Todd directions. In this article we show that these direction fields are special cases of so-called target directions, a unifying concept for primal-dual interior point methods for self-scaled conic programming. In particular, this implies that NesterovTodd directions derive from a Newton system.