A review of the LU update in the simplex algorithm

In the simplex method for linear optimisation, two linear systems involving a square non-singular basis matrix B and its transpose are solved at each iteration. A column of B is then replaced, and the solution of these two linear systems is required again. For the past four decades, researchers have made considerable advancements in updating the sparse LU factors of the basis matrix in this algorithm. This paper reviews the state-of-the-art in LU update procedures, while analysing each procedure to gain a full understanding of the progress made in this area.

[1]  Fuzhen Zhang.  The Schur complement and its applications , 2005 .

[2]  Uwe H. Suhl,et al.  Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms , 2007, Comput. Optim. Appl..

[3]  Robert E. Bixby,et al.  MIP: Theory and Practice - Closing the Gap , 1999, System Modelling and Optimization.

[4]  Achim Koberstein,et al.  Progress in the dual simplex algorithm for solving large scale LP problems: techniques for a fast and stable implementation , 2008, Comput. Optim. Appl..

[5]  William Orchard-Hays,et al.  Advanced Linear-Programming Computing Techniques , 1968 .

[6]  Uwe H. Suhl,et al.  A fast LU update for linear programming , 1993, Ann. Oper. Res..

[7]  Richard D. McBride.  A bump triangular dynamic factorization algorithm for the simplex method , 1980, Math. Program..

[8]  Pradeep Dubey,et al.  Larrabee: A Many-Core x86 Architecture for Visual Computing , 2009, IEEE Micro.

[9]  K. I. M. McKinnon,et al.  Hyper-Sparsity in the Revised Simplex Method and How to Exploit it , 2005, Comput. Optim. Appl..

[10]  P. Gill,et al.  Sparse Matrix Methods in Optimization , 1984 .

[11]  David Patterson,et al.  The Top 10 Innovations in the New NVIDIA Fermi Architecture, and the Top 3 Next Challenges , 2009 .

[12]  Michael A. Saunders,et al.  A Block-LU Update for Large-Scale Linear Programming , 1992, SIAM J. Matrix Anal. Appl..

[13]  John K. Reid,et al.  A sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases , 1982, Math. Program..

[14]  S. K. Eldersveld,et al.  Vectorization algorithm for the solution of large, sparse triangular systems of equations. Technical report , 1990 .

[15]  H. Huynh.  A Large-Scale Quadratic Programming Solver Based on Block-Lu Updates of the KKT System , 2008 .

[16]  Roy E. Marsten,et al.  The Design of the XMP Linear Programming Library , 1981, TOMS.

[17]  Nicholas I. M. Gould,et al.  Preprocessing for quadratic programming , 2004, Math. Program..

[18]  Eli Hellerman,et al.  Reinversion with the preassigned pivot procedure , 1971, Math. Program..

[19]  Robert E. Bixby,et al.  Solving Real-World Linear Programs: A Decade and More of Progress , 2002, Oper. Res..

[20]  J. Gilbert,et al.  Sparse Partial Pivoting in Time Proportional to Arithmetic Operations , 1986 .

[21]  John A. Tomlin,et al.  Updated triangular factors of the basis to maintain sparsity in the product form simplex method , 1972, Math. Program..

[22]  Roger Fletcher,et al.  Stable modification of explicitLU factors for simplex updates , 1984, Math. Program..

[23]  Roger Espasa.  Larrabee: A Many-Core Intel Architecture for Visual Computing , 2010, HiPEAC.

[24]  Alexander Meeraus,et al.  Matrix augmentation and partitioning in the updating of the basis inverse , 1977, Math. Program..

[25]  Nikolaos V. Sahinidis,et al.  Computational experience with linear optimization and related problems , 2010 .

[26]  Alexander Meeraus,et al.  Matrix augmentation and structure preservation in linearly constrained control problems , 1980, Math. Program..

[27]  P. Gill,et al.  Maintaining LU factors of a general sparse matrix , 1987 .

[28]  Nikolaos V. Sahinidis,et al.  A review of LU factorisation in the simplex algorithm , 2012, Int. J. Math. Oper. Res..

[29]  J. A. Tomlin.  An accuracy test for updating triangular factors , 1975 .

[30]  André F. Perold,et al.  A degeneracy exploiting LU factorization for the simplex method , 1980, Math. Program..

[31]  Samuel Williams,et al.  The potential of the cell processor for scientific computing , 2005, CF '06.

[32]  J. Tomlin.  Pivoting for Size and Sparsity in Linear Programming Inversion Routes , 1972 .

[33]  R. Willoughby,et al.  Some results on sparse matrices , 1970 .

[34]  P. E. Proctor.  Implementation of a Double-Basis Simplex Method for the General Linear Programming Problem , 1985 .

[35]  I. Maros.  Computational Techniques of the Simplex Method , 2002 .

[36]  Gene H. Golub,et al.  The simplex method of linear programming using LU decomposition , 1969, Commun. ACM.

[37]  H. Peter Hofstee,et al.  Key features of the design methodology enabling a multi-core SoC implementation of a first-generation CELL processor , 2006, Asia and South Pacific Conference on Design Automation, 2006..