Interior point algorithms for linear programming achieve significant reductions in computer time over earlier methods for many large linear programming problems (LPs) and solve problems larger than previously possible. The most computationally intensive step in each iteration of any interior point algorithm is the numerical factorization of a sparse, symmetric, positive definite matrix. In large or relatively dense problems, 80–90% or more of computational time is spent in this step. This study describes our implementations of two algorithms for performing this factorization, the column Cholesky and the multifrontal methods, based on graph theory applied to sparse symmetric matrices. We use advanced techniques such as loop unrolling and equivalent sparse matrix reordering to improve the performance of the factorization step. Our studies are incorporated into an implementation of the primal-dual barrier algorithm. Computational experiments on relatively large LPs on a DEC station 3100 demonstrate that the primal-dual barrier algorithm using our advanced column Cholesky outperforms by 20–60% the same algorithm using a straightforward column Cholesky. Also, our multifrontal method using the loop unrolling technique in a partial inner product routine shows a 10–50% speedup compared with no loop unrolling. The two methods exhibit comparable overall performance. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A survey of the significant developments in the field of interior point methods for linear programming is presented, beginning with Karmarkar's projective algorithm and concentrating on the many variants that can be derived from logarithmic barrier methods. Full implementation details of the primal-dual predictor-corrector code OB1 are given, including preprocessing, matrix orderings, and matrix factorization techniques. A computational comparison of OB1 with a state-of-the-art simplex code using eight large models is given. In addition, computational results are presented where OB1 is used to solve two very large models that have never been solved by any simplex code INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Rejoinder to commentaries to the author's feature article “Interior Point Methods for Linear Programming: Computational State of art.” INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

We report tests of CPLP, a piecewise-linear simplex implementation based on the XMP subroutine library. Using test problems of various shapes and origins, we judge CPLP against a comparable linear bounded-variable simplex code that is applied to equivalent linear programs. Our evidence indicates that CPLP is typically 2 to 3 times faster in solving the test problems that are most significantly piecewise-linear; previously predicted advantages of the piecewise-linear simplex approach are observed to occur, though with different combinations of advantages being relevant in different cases. The savings achieved by CPLP appear likely to extend, moreover, to the costs involved in formulating, generating and analyzing piecewise-linear programs. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

The impact of incorporating a specific concurrency control mechanism (CCM) into the file allocation problem (FAP) is discussed. Depending on the specific CCM use, the communication flows in a network will vary. To allocate data optimally, one must identify the exact communication flows in the network. It is this aspect that has been ignored in past research on the FAP. A linear mixed-integer programming model formulated for the FAP is given. The model incorporates the WRITE LOCKS ALL-READ LOCKS ONE mechanism for concurrency control. A special algorithm based on the implicit representation of variable upper bounds is developed to solve the model. Detailed analysis for various configurations of a network is performed. Several potential applications for the model are identified.>

The dual affine interior point method is extended to handle variables with simple upper bounds as well as free variables. During execution, variables which appear to be going to zero are fixed at zero, and rows with slack variables bounded away from zero are removed. A variant of the big-M artificial variable method to attain feasibility is derived. The simplex method is used to recover an optimal basis upon completion of the algorithm, and the effects of scaling are discussed. Computational experience on a variety of problems is presented. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A branch-and-bound algorithm for mixed-integer programming (MIP) is presented in which a different integer variable is designated for branching at each depth in the search tree. This is done in order to achieve economies of scale in solving the linear programming relaxations to obtain bounds for the (MIP) algorithm. This algorithm is the foundation for the zoom software system, which is briefly described. Illustrative computational experience is included. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Abstract The set partitioning model of the crew rotation problem has been well known for many years. This paper focuses on recent developments that have made this model more attractive and have resulted in several successful implementations. These developments include improved problem conceptualizations and decompositions, as well as new solution techniques employing Lagrangian relaxation and subgradient optimization. Experience is reported from The Flying Tiger Line, Pacific Southwest Airways, Continental Airlines, and Helsinki City Transport. A case is made for work on heuristic decomposition methods to break large problems into moderate sized pieces that can be solved exactly.

article Free Access Share on The Design of the XMP Linear Programming Library Author: Roy E. Marsten Department of Management Information Systems, College of Business and Public Administration, University of Arizona, Tucson, AZ Department of Management Information Systems, College of Business and Public Administration, University of Arizona, Tucson, AZView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 7Issue 4Dec. 1981 pp 481–497https://doi.org/10.1145/355972.355976Published:01 December 1981Publication History 206citation566DownloadsMetricsTotal Citations206Total Downloads566Last 12 Months34Last 6 weeks8 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF