Jacek Gondzio

Multiple centrality corrections in a primal-dual method for linear programming

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.

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of todays codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.

Solving nonlinear multicommodity flow problems by the analytic center cutting plane method

The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with Dijkstra’s d-heap algorithm. An implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities).

HOPDM (version 2.12) — A fast LP solver based on a primal-dual interior point method☆

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 todays commercial LP solvers.

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.

Regularized Symmetric Indefinite Systems in Interior Point Methods for Linear and Quadratic Optimization

This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraint. The new regularization techniques for Newton equation system applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidefinite systems known to be strongly factorizable to a form of Cholesky-like factorization.

Warm start of the primal-dual method applied in the cutting-plane scheme

A practical warm-start procedure is described for the infeasible primal-dual interior-point method (IPM) 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.

Parallel Interior Point Solver for Structured Linear Programs

Abstract. Issues of implementation of an object-oriented library for parallel interior-point methods are addressed. The solver can easily exploit any special structure of the underlying optimization problem. In particular, it allows a nested embedding of structures and by this means very complicated real-life optimization problems can be modelled. The efficiency of the solver is illustrated on several problems arising in the optimization of networks. The sequential implementation outperforms the state-of-the-art commercial optimization software. The parallel implementation achieves speed-ups of about 3.1-3.9 on 4-processors parallel systems and speed-ups of about 10-12 on 16-processors parallel systems.

Base Station Location Optimization for Minimal Energy Consumption in Wireless Networks

This paper studies the combined problem of base station location and optimal power allocation, in order to optimize the energy efficiency of a cellular wireless network. Recent work has suggested that moving from a network of a small number of high power macrocells to a larger number of smaller microcells may improve the energy efficiency of the network. This paper investigates techniques to optimize the number of base stations and their locations, in order to minimize energy consumption. An important contribution of the paper is that it takes into account non-uniform user distributions across the coverage area, which is likely to be encountered in practice. The problem is solved using approaches from optimization theory that deal with the facility location problem. Stochastic programming techniques are used to deal with the expected user distributions. An example scenario is presented to illustrate how the technique works and the potential performance gains that can be achieved.

Reoptimization With the Primal-Dual Interior Point Method

Reoptimization techniques for an interior point method applied to solving a sequence of linear programming problems are discussed. Conditions are given for problem perturbations that can be absorbed in merely one Newton step. The analysis is performed for both short-step and long-step feasible path-following methods. A practical procedure is then derived for an infeasible path-following method. It is applied in the context of crash start for several large-scale structured linear programs. Numerical results with OOPS, a new object-oriented parallel solver, demonstrate the efficiency of the approach. For large structured linear programs, crash start leads to about 40% reduction in the number of iterations and translates into a 25% reduction of the solution time. The crash procedure parallelizes well, and speed-ups between 3.1--3.8 on four processors are achieved.