Csaba Mészáros

Implementation of Interior-Point Methods for Large Scale Linear Programs

In this paper we give an overview of the mostimportant characteristics of advanced implementations of interior point methods.

A repository of convex quadratic programming problems

The introduction of a standard set of linear programming problems, to be found in NETLIB/-LP/DATA, had an important impact on measuring, comparing and reporting the performance of LP solvers. Until recently the efficiency of new algorithmic developments has been measured using this important reference set. Presently, we are witnessing an ever growing interest in the area of quadratic programming. The research community is somewhat troubled by the lack of a standard format for defining a QP problem and also by the lack of a standard reference set of problems for purposes similar to that of LP. In the paper we propose a standard format and announce the availability of a test set of collected 138 QP problems.

Fast Cholesky factorization for interior point methods of linear programming

Abstract Every iteration of an interior point method of large scale linear programming requires computing at least one orthogonal projection. In practice, Cholesky decomposition seems to be the most efficient and sufficiently stable method. We studied the ‘column oriented’ or ‘left looking’ sparse variant of the Cholesky decomposition, which is a very popular method in large scale optimization. We show some techniques such as using supernodes and loop unrolling for improving the speed of computation. We show numerical results on a wide variety of large scale, real-life linear programming problems.

The BPMPD interior point solver for convex quadratic problems

The paper describes the convex quadratic solver BPMPD Version 2.21. The solver is based on the infeasible–primal–dual algorithm extended by the predictor–corrector and target–following techniques. The discussion includes topics related to the implemented algorithm and numerical algebra employed. We outline the presolve, scaling and starting point stategies used in BPMPD, and special attention is given for sparsity and stability issues. Computational results are given on a demonstrative set of convex quadratic problems.

On sensitivity analysis for a class of decision systems

The sensitivity analysis of general decision systems originated from the Bridgman model is dealt with. It is shown that these problems are equivalent to the optimization of linear fractional functions over rectangles. By using the specialities of decision problems, an O(n log n) algorithm is elaborated fro solving such problems. Computational experience is also given.

The role of the augmented system in interior point methods

Abstract We present a way to use the augmented system approach in interior point methods. We elaborate on the increased freedom in determining the pivot order which makes this approach computationally very competitive. This means that with the pivot search heuristics presented here, in most of the cases we can achieve a performance not worse than the AD −1 A T method, and in several cases much better. In reality, the augmented system seems to be the only safe method in case of dense columns or ‘bad’ nonzero pattern. We found that both methods are important for their usefulness. Our implementation includes both. It is also equipped with an analyzer that is able to determine which of them to use. It is based on the evaluation of the nonzero pattern of the constraint matrix. We also point out that the treatment of free variables is also more efficient in the framework of the augmented system. We report on some very favorable computational experiences achieved with our implementation of the augmented system based on these ideas.

On free variables in interior point methods

Interior point methods, especially the algorithms for linear programming problems are sensitive if there are unconstrained (free) variables in the problem. While replacing a free variable by two nonnegative ones may cause numerical instabilities, the implicit handling results in a semidefinite scaling matrix at each interior point iteration. In the paper we investigate the effects if the scaling matrix is regularized. Our analysis will prove that the effect of the regularization can be easily monitored and corrected if necessary. We describe the regularization scheme mainly for the efficient handling of free variables, but a similar analysis can be made for the case, when the small scaling factors are raised to larger values to improve the numerical stability of the systems that define the searcn direction. We will show the superiority of our approach over the variable replacement method on a set of test problems arising from water management application

The augmented system variant of IPMs in two-stage stochastic linear programming computation

Abstract The application of interior point methods (IPM) to solve the deterministic equivalent of two-stage stochastic linear programming problems is a known and natural idea. Experiments have proved that among the interior point methods, the augmented system approach gives the best performance on these problems. However, most of their implementations encounter numerical difficulties in certain cases, which can result in loss of efficiency. We present a new approach for the decomposition of the augmented system, which ‘automatically’ exploits the special behavior of the problems. We show that the suggested approach can be implemented in a fast and numerically robust way by solving a number of large-scale two-stage stochastic linear programming problems. The comparison of our solver with fo1aug, which is considered as a state-of-the-art augmented system implementation of interior point methods, is also given.

On Numerical Issues of Interior Point Methods

This paper concerns some numerical stability issues of factorizations in interior point methods. In our investigation we focus on regularization techniques for the augmented system. We derive the fundamental property of regularization and necessary conditions for the convergence of iterative refinement. A relaxation technique is described that improves on convergence properties. We introduce a practical, adaptive technique to determine the required amount of regularization in numerically difficult situations. Numerical experiments on large-scale, numerically difficult linear programming problems are presented.

A flexible framework for Group Decision Support: WINGDSS version 3.0

A flexible framework for Group Decision Support on PCs in Microsoft Windows environment is presented. WINGDSS does not replace human judgment but highly supports the entire process of decision making, from problem structuring to post-decision analysis. The latest version of WINGDSS is a modular, open system with a dynamical connection to relational databases, an interpreter for defining problem specific evaluation procedures, a lot of interactive features from setting up the tree of criteria until the sensitivity analysis on individual/group ranking. By providing tools for recursively redefining the decision problem, WINGDSS helps the decision makers in achieving a result satisfactory to the whole group.