Uwe H. Suhl

Computing Sparse LU Factorizations for Large-Scale Linear Programming Bases

This paper discusses the computation of LU factorizations for large sparse matrices with emphasis on large-scale linear programming bases. We present new implementation techniques which reduce the computation times significantly. Numerical experiments with large-scale real life test problems were conducted. The software is compared with the basis factorization of MPSX/370, IBMs commercial LP system. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A fast LU update for linear programming

This paper discusses sparse matrix kernels of simplex-based linear programming software. State-of-the-art implementations of the simplex method maintain an LU factorization of the basis matrix which is updated at each iteration. The LU factorization is used to solve two sparse sets of linear equations at each iteration. We present new implementation techniques for a modified Forrest-Tomlin LU update which reduce the time complexity of the update and the solution of the associated sparse linear systems. We present numerical results on Netlib and other real-life LP models.

MOPS — Mathematical optimization system

Abstract This paper discusses a software system for solving large scale linear and mixed-integer optimization models. The system in its present form is the result of several years of research and development. The first functional version of MOPS was completed 1987 for the IBM 6150 (a predecessor of the RS/6000) under AIX. A benchmark model from the Petroleum industry with 5563 rows and 6181 columns was solved with MOPS in 5 hours. The newest version solves this model in 6.51 minutes on a MS-DOS PC with i80486 processor (25 Mhz). This paper discusses only LP optimization. The MIP optimizer in MOPS is a prototype version which will undergo major changes in the future. We describe algorithmic aspects, implementation issues and numerical results on Netlib and other real-life LP models. Sophisticated LP optimizers — although simple from a mathematical point of view — are algorithmically very complex. The discussion is therefore limited to a few fundamental issues that we feel are important.

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

Abstract The dual simplex algorithm has become a strong contender in solving large scale LP problems. One key problem of any dual simplex algorithm is to obtain a dual feasible basis as a starting point. We give an overview of methods which have been proposed in the literature and present new stable and efficient ways to combine them within a state-of-the-art optimization system for solving real world linear and mixed integer programs. Furthermore, we address implementation aspects and the connection between dual feasibility and LP-preprocessing. Computational results are given for a large set of large scale LP problems, which show our dual simplex implementation to be superior to the best existing research and open-source codes and competitive to the leading commercial code on many of our most difficult problem instances.

Supernode processing of mixed-integer models

This paper discusses processing software for large scale mixed-integer optimization models. The software is part of the Mathematical OPtimization System MOPS [18] which contains algorithms for solving large-scale LP and mixed-integer programs. The processing techniques are implemented in such a way that they can be applied not only initially but also during the branch-and-bound algorithm.This paper discusses only a subset of the processing techniques included in MOPS. Algorithmic and software design aspects of the branch-and-bound process are not part of this paper.

A branch‐and‐cut algorithm for solving generalized multiperiod Steiner problems in graphs

Given is an undirected graph with positive or negative edge weights which represent a profit if an investment such as installing a gas pipe takes place in a given time period. A certain part of the graph may already be piped in previous periods. The task is to extend the piped subgraph in the most profitable way over a multiperiod time horizon. In addition, there may be general side constraints limiting the extensions per period. This problem is a generalization of the Steiner problem in graphs. It is shown that the general Steiner problem in graphs is equivalent to the node-weighted Steiner problem in graphs. A branch-and-cut algorithm is presented which is based on an integer programming formulation to solve the multiperiod Steiner problem in graphs. Numerical results with real-life problems of a gas utility company show that optimal solutions can be obtained in minutes of computer time on a fast PC.

Design and Implementation of a fast and portable LP-Optimizer

We describe scope, design and implementation aspects of an LP-optimizer to be called MOPS (Mathematical Optimization System) which was developed on an IBM 6150 workstation under AIX (a UNIX derivative). The system can be characterized as follows: high performance by using the most advanced algorithms for solving large-scale LP-problems and an efficient implementation of the algorithms. Recent results in sparse matrix research are used. On powerful workstations or mainframes MOPS allows the solution of large-scale optimization problems. On PC’s with DOS problem size is limited by the amount of memory available. Compared to other university developed LP-codes MOPS is designed for solving very large and difficult problems.

Finding Feasible Solutions to Hard Mixed-integer Programming Problems Using Hybrid Heuristics

In current mixed-integer programming (MIP) solvers heuristics are used to find feasible solutions before the branch-and-bound or branchand-cut algorithm is applied to the problem. Knowing a feasible solution can improve the solutions found or the time to solve the problem very much. This paper discusses hybrid heuristics for this purpose. Hybrid in this context means that these heuristics use the branch-and-bound algorithm to search a smaller subproblem. Several possible hybrid heuristics are presented and computational results are given.

Implementing an Algorithm: Performance Considerations and a Case Study

After a general discussion of program performance and its influence factors we consider a case study. A specific version of TOYODA’s heuristic algorithms for solving multidimensional knapsack problems was implemented in three different FORTRAN programs. Starting from a straightforward program, we introduce increasingly sophisticated data structures to improve the virtual execution time of the programs. All programs were compiled with the enhanced version of the FORTRAN H compiler available on IBM/370, 43xx, 303x machines, using its four optimization options. A computational study was performed with a series of randomly generated test problems with up to 3200 variables and constraints and about 70000 nonzeros. The purpose of this study was to measure virtual central processor times and working set sizes as a function of problem size and to study the influence of code optimization performed by the compiler. Since all three programs are representations of the same algorithm, conclusions can be drawn as to the relative importance of the influence factors on program performance. As is also demonstrated, the potential performance of a representation of an abstract algorithm may be difficult to project without a careful implementation.

Solving Airline-fleet Scheduling Problems with Mixed-integer Programming

This chapter discusses the solution of real-life airline fleet scheduling problems of a large European airline, and aims to minimize the number of aircraft needed to serve a given set of flights. There is certain Nfreedom to schedule the flights expressed as a time window per each flight within which the flight has to depart. This is the strategy of some European carriers, which first fix the number of flights per connection together with the time window and the fleet for each flight according to the expected number of passengers. In a second step they schedule the flights within their given departure time windows. This problem of scheduling the flights and simultaneously generating aircraft rotations will be called the fleet scheduling problem. (For more details on the planning process see Suhl, 1995)