Implementation of the Simplex Method

All algorithms formulated in this book, such as the simplex algorithm and the dual simplex algorithm, are theoretical or conceptional, and can not be put to practical use via programming directly. Softwares, resulting by following the algorithms, step by step naively, would only solve textbook instances, involving only few variables and constraints, not real-world problems, especially large-scale sparse problems. Experiences indicate that implementation details are crucial to algorithms’ success (Orchard-Hays 1954; Tomlin 1974; Todd 1982, 1983; Bixby 1994).

[1]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[2]  Pablo Guerrero-García,et al.  A deficient-basis dual counterpart of Paparrizos, Samaras and Stephanides’ primal-dual simplex-type algorithm , 2009, Optim. Methods Softw..

[3]  R. Bartels.  A stabilization of the simplex method , 1971 .

[4]  V. Chvátal,et al.  Notes on Bland’s pivoting rule , 1978 .

[5]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956 .

[6]  Wei Li.  A note on two direct methods in linear programming , 2004, Eur. J. Oper. Res..

[7]  J. Maitland,et al.  Operational Research and Advertising: Theories of Response , 1960 .

[8]  George L. Nemhauser,et al.  The Age of Optimization: Solving Large-Scale Real-World Problems , 1994, Oper. Res..

[9]  J. Tomlin.  An experimental approach to karmarkar’s projective method for linear programming , 1987 .

[10]  Hamdy A. Taha,et al.  Integer Programming: Theory, Applications, and Computations , 1975 .

[11]  Pingqi Pan.  A FAST SIMPLEX ALGORITHM FOR LINEAR PROGRAMMING , 2010 .

[12]  Michael A. Saunders,et al.  A FAST, STABLE IMPLEMENTATION OF THE SIMPLEX METHOD USING BARTELS-GOLUB UPDATING , 1976 .

[13]  Michel Bénichou,et al.  The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results , 1977, Math. Program..

[14]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.

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

[16]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[17]  R. Willoughby.  Sparse matrices and their applications , 1972 .

[18]  J. Farkas.  Theorie der einfachen Ungleichungen. , 1902 .

[19]  T. L. Saaty,et al.  The computational algorithm for the parametric objective function , 1955 .

[20]  Harvey J. Greenberg.  A Tutorial on Matricial Packing , 1978 .

[21]  Shu-Cherng Fang,et al.  Linear Optimization and Extensions: Theory and Algorithms , 1993 .

[22]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .

[23]  Donald Goldfarb,et al.  On the Bartels—Golub decomposition for linear programming bases , 1977, Math. Program..

[24]  G. B. Dantzig,et al.  A strictly improving linear programming Phase I algorithm , 1993, Ann. Oper. Res..

[25]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[26]  P. Wolfe.  The Composite Simplex Algorithm , 1965 .

[27]  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..

[28]  J. W. Suurballe,et al.  A note on cycling in the simplex method , 1969 .

[29]  Ping-Qi Pan,et al.  A projective simplex method for linear programming , 1999 .

[30]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[31]  Nimrod Megiddo,et al.  Progress in Mathematical Programming , 1989 .

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

[33]  Nicholas I. M. Gould,et al.  New crash procedures for large systems of linear constraints , 1989, Math. Program..

[34]  Zhi-Quan Luo,et al.  A modified predictor-corrector method for linear programming , 1994, Comput. Optim. Appl..

[35]  Pan,et al.  A NEW PERTURBATION SIMPLEX ALGORITHM FOR LINEAR PROGRAMMING , 1999 .

[36]  Robert J. Vanderbei,et al.  Two-thirds is sharp for affine scaling , 1993, Oper. Res. Lett..

[37]  O. H. Brownlee,et al.  ACTIVITY ANALYSIS OF PRODUCTION AND ALLOCATION , 1952 .

[38]  Nimrod Megiddo,et al.  A note on degeneracy in linear programming , 1986, Math. Program..

[39]  Artur Swietanowski,et al.  A New Steepest Edge Approximation for the Simplex Method for Linear Programming , 1998, Comput. Optim. Appl..

[40]  Robert G. Bland,et al.  New Finite Pivoting Rules for the Simplex Method , 1977, Math. Oper. Res..

[41]  David I. Steinberg,et al.  Letter to the Editor - On the Possibility of Cycling with the Simplex Method , 1978, Oper. Res..

[42]  K. Kortanek,et al.  Convergence results and numerical experiments on a linear programming hybrid algorithm , 1987 .

[43]  M. C. Cheng.  General criteria for redundant and nonredundant linear inequalities , 1987 .

[44]  Richard P. Brent,et al.  The Complexity of computational problem solving , 1976 .

[45]  Richard W. Cottle,et al.  Least-Index Resolution of Degeneracy in Linear Complementarity Problems with Sufficient Matrices , 1979, SIAM J. Matrix Anal. Appl..

[46]  Michael A. Malcolm,et al.  Computer methods for mathematical computations , 1977 .

[47]  Roy E. Marsten,et al.  On Implementing Mehrotra's Predictor-Corrector Interior-Point Method for Linear Programming , 1992, SIAM J. Optim..

[48]  Roy E. Marsten,et al.  Feature Article - Interior Point Methods for Linear Programming: Computational State of the Art , 1994, INFORMS J. Comput..

[49]  A. J. Hoffman.  Cycling in the Simplex Algorithm , 2003 .

[50]  George B. Dantzig,et al.  7* A Primal-Dual Algorithm for Linear Programs , 1957 .

[51]  Gautam Mitra,et al.  Analysis of mathematical programming problems prior to applying the simplex algorithm , 1975, Math. Program..

[52]  G. Dantzig.  Programming of Interdependent Activities: II Mathematical Model , 1949 .

[53]  Pingqi Pan.  A Basis-Deficiency-Allowing Variation of the Simplex Method for Linear Programming , 1998 .

[54]  M. Kojima,et al.  A primal-dual interior point algorithm for linear programming , 1988 .

[55]  T Talaky,et al.  Interior Point Methods of Mathematical Programming , 1997 .

[56]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[57]  P. Gill,et al.  A numerically stable form of the simplex algorithm , 1973 .

[58]  Bob Hattersley,et al.  A dual approach to primal degeneracy , 1988, Math. Program..

[59]  Jeffery L. Kennington,et al.  An Empirical Evaluation of the KORBX® Algorithms for Military Airlift Applications , 1990, Oper. Res..

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

[61]  Kees Roos,et al.  An exponential example for Terlaky's pivoting rule for the criss-cross simplex method , 1990, Math. Program..

[62]  Gene H. Golub,et al.  Numerical methods for solving linear least squares problems , 1965, Milestones in Matrix Computation.

[63]  Ping-Qi Pan.  A primal deficient-basis simplex algorithm for linear programming , 2008, Appl. Math. Comput..

[64]  R. Tapia,et al.  An Optimal Basis Identification Technique for Interior-Point Linear Programming Algorithms , 1991 .

[65]  Nimrod Megiddo,et al.  Boundary Behavior of Interior Point Algorithms in Linear Programming , 1989, Math. Oper. Res..

[66]  L. Hurwicz,et al.  Reduction of Constrained Maxima to Saddle-point Problems , 2014 .

[67]  C. C. Gonzaga,et al.  An Algorithm for Solving Linear Programming Problems in O(n 3 L) Operations , 1989 .

[68]  Ping-Qi Pan.  The most-obtuse-angle row pivot rule for achieving dual feasibility: A computational study , 1997 .

[69]  George B. Dantzig,et al.  Decomposition Principle for Linear Programs , 1960 .

[70]  Robert G. Jeroslow,et al.  The simplex algorithm with the pivot rule of maximizing criterion improvement , 1973, Discret. Math..

[71]  C. E. Lemke,et al.  The dual method of solving the linear programming problem , 1954 .

[72]  S. Gass.  George B. Dantzig , 2011 .

[74]  Harvey J. Greenberg,et al.  An exact update for Harris’ TREAD , 1975 .

[75]  P. Wolfe.  A Technique for Resolving Degeneracy in Linear Programming , 1963 .

[76]  C. Botsaris.  Differential gradient methods , 1978 .

[77]  A. W. Tucker,et al.  Duality Theory of Linear Programs: A Constructive Approach with Applications , 1969 .

[78]  J. F. Benders.  Partitioning procedures for solving mixed-variables programming problems , 1962 .

[79]  N. Megiddo.  Pathways to the optimal set in linear programming , 1989 .

[80]  Ping-Qi Pan,et al.  Efficient nested pricing in the simplex algorithm , 2008, Oper. Res. Lett..

[81]  Bahman Kalantari,et al.  Karmarkar's algorithm with improved steps , 1990, Math. Program..

[82]  M. C. Bartholomew-Biggs,et al.  ODE versus SQP methods for constrained optimization , 1989 .

[83]  Włodzimierz Ogryczak.  The simplex method is not always well behaved , 1988 .

[84]  István Maros.  A general Phase-I method in linear programming☆ , 1983 .

[85]  J. Stoer,et al.  A Realization of the Simplex Method Based on Triangular Decompositions , 1971 .

[86]  R. Bellman,et al.  Linear Programming and Economic Analysis. , 1960 .

[87]  D. M. Ryan,et al.  On the solution of highly degenerate linear programmes , 1988, Math. Program..

[88]  E. M. L. Beale,et al.  An alternative method for linear programming , 1954, Mathematical Proceedings of the Cambridge Philosophical Society.

[89]  M. Saunders.  Large-scale linear programming using the Cholesky factorization , 1972 .

[90]  T. Chan.  On the existence and computation of $LU$-factorizations with small pivots , 1984 .

[91]  Wei Li,et al.  Combined projected gradient algorithm for linear programming , 2006, Optim. Methods Softw..

[92]  Michael A. Saunders,et al.  On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method , 1986, Math. Program..

[93]  Kees Roos,et al.  A survey of search directions in interior point methods for linear programming , 1991, Math. Program..

[94]  Robert E. Bixby,et al.  Progress in Linear Programming , 1993 .

[95]  A. M. Geoffrion.  Generalized Benders decomposition , 1972 .

[96]  Michael J. Todd,et al.  Large-scale linear programming: Geometry, working bases and factorizations , 1983, Math. Program..

[97]  Walter F. Mascarenhas,et al.  The Affine Scaling Algorithm Fails for Stepsize 0.999 , 1997, SIAM J. Optim..

[98]  Y. Ye.  Eliminating columns in the simplex method for linear programming , 1989 .

[99]  E. M. L. Beale,et al.  Mathematical programming in practice , 1968 .

[100]  William Orchard Hayes.  EVOLUTION OF COMPUTER CODES FOR LINEAR PROGRAMMING , 1956 .

[101]  Richard W. Hamming,et al.  Introduction to Applied Numerical Analysis. , 1971 .

[102]  J. H. Wilkinson.  Modern Error Analysis , 1971 .

[103]  John A. Tomlin,et al.  On pricing and backward transformation in linear programming , 1974, Math. Program..

[104]  Uwe H. Suhl,et al.  MOPS -- Mathematical optimization system , 1994 .

[105]  Masakazu Muramatsu,et al.  Global Convergence of a Long-Step Affine Scaling Algorithm for Degenerate Linear Programming Problems , 1995, SIAM J. Optim..

[106]  Yinyu Ye,et al.  Convergence behavior of interior-point algorithms , 1993, Math. Program..

[107]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part I: Linear programming , 1989, Math. Program..

[108]  Rao R. Vemuganti.  On gradient simplex methods for linear programs , 2004, Adv. Decis. Sci..

[109]  Z. Zlatev.  On Some Pivotal Strategies in Gaussian Elimination by Sparse Technique , 1980 .

[110]  R. Bixby,et al.  A note on detecting simple redundancies in linear systems , 1987 .

[111]  Steve Weal.  History of Mathematical Programming , 1992 .

[112]  G. Zoutendijk,et al.  Methods of Feasible Directions , 1962, The Mathematical Gazette.

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

[114]  Martin Grötschel,et al.  Mathematical Programming The State of the Art, XIth International Symposium on Mathematical Programming, Bonn, Germany, August 23-27, 1982 , 1983, ISMP.

[115]  Ángel Santos-Palomo,et al.  The sagitta method for solving linear programs , 2004, Eur. J. Oper. Res..

[116]  M. J. D. Powell,et al.  A tolerant algorithm for linearly constrained optimization calculations , 1989, Math. Program..

[117]  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..

[118]  Jian-Feng Hu,et al.  An efficient approach to updating simplex multipliers in the simplex algorithm , 2008, Math. Program..

[119]  Michael A. Saunders,et al.  Large-scale linearly constrained optimization , 1978, Math. Program..

[120]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[121]  Ping-Qi Pan,et al.  An affine-scaling pivot algorithm for linear programming , 2013 .

[122]  J. Descloux,et al.  Dégénérescence dans les approximations de Tschebyscheff linéaires et discrètes , 1961 .

[123]  Ping-Qi Pan,et al.  A dual projective simplex method for linear programming , 1998 .

[124]  Michael J. Todd,et al.  An implementation of the simplex method for linear programming problems with variable upper bounds , 1982, Math. Program..

[125]  Jean-Philippe Vial,et al.  Long-step primal-dual target-following algorithms for linear programming , 1996, Math. Methods Oper. Res..

[126]  Kurt M. Anstreicher,et al.  A Family of Search Directions for Karmarkar's Algorithm , 1993, Oper. Res..

[127]  G. Stewart,et al.  Sparse Matrix Proceedings. , 1980 .

[128]  Jian-Feng Hu,et al.  A note on "an improved initial basis for the simplex algorithm" , 2007, Comput. Oper. Res..

[129]  A. Charnes.  Optimality and Degeneracy in Linear Programming , 1952 .

[130]  István Maros.  A generalized dual phase-2 simplex algorithm , 2003, Eur. J. Oper. Res..

[131]  Clyde L. Monma,et al.  An Implementation of a Primal-Dual Interior Point Method for Linear Programming , 1989, INFORMS J. Comput..

[132]  I. Lustig,et al.  Computational experience with a primal-dual interior point method for linear programming , 1991 .

[133]  Robert E. Bixby,et al.  Implementing the Simplex Method: The Initial Basis , 1992, INFORMS J. Comput..

[134]  K. Turner.  Computing projections for the Karmarkar algorithm , 1991 .

[135]  Robert E. Bixby,et al.  Commentary - Progress in Linear Programming , 1994, INFORMS J. Comput..

[136]  G. Dantzig,et al.  THE PRODUCT FORM FOR THE INVERSE IN THE SIMPLEX METHOD , 1954 .

[137]  Uwe H. Suhl,et al.  Computing Sparse LU Factorizations for Large-Scale Linear Programming Bases , 1990, INFORMS J. Comput..

[138]  R. Kipp Martin,et al.  Large scale linear and integer optimization - a unified approach , 1998 .

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

[140]  Takashi Tsuchiya,et al.  Global Convergence Property of the Affine Scaling Methods for Primal Degenerate Linear Programming Problems , 1992, Math. Oper. Res..

[141]  V. Klee.  A class of linear programming problems requiring a large number of iterations , 1965 .

[142]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[143]  Stanley Zionts,et al.  The Criss-Cross Method for Solving Linear Programming Problems , 1969 .

[144]  Sanjay Mehrotra.  On finding a vertex solution using interior point methods , 1991 .

[145]  Yinyu Ye,et al.  A generalized homogeneous and self-dual algorithm for linear programming , 1995, Oper. Res. Lett..

[146]  Jean-Philippe Vial,et al.  A polynomial method of approximate centers for linear programming , 1992, Math. Program..

[147]  Yinyu Ye,et al.  A “build-down” scheme for linear programming , 1990, Math. Program..

[148]  Ping-Qi Pan,et al.  A largest-distance pivot rule for the simplex algorithm , 2008, Eur. J. Oper. Res..

[149]  Harvey J. Greenberg,et al.  Pivot Selection Tactics , 1978 .

[150]  D. Shanno,et al.  A unified view of interior point methods for linear programming , 1990 .

[151]  J. Reid,et al.  On the Automatic Scaling of Matrices for Gaussian Elimination , 1972 .

[152]  Donald Goldfarb,et al.  A practicable steepest-edge simplex algorithm , 1977, Math. Program..

[153]  Eli Hellerman,et al.  The Partitioned Preassigned Pivot Procedure (P4) , 1972 .

[154]  Ekaterina A. Kostina.  The long step rule in the bounded-variable dual simplex method: Numerical experiments , 2002, Math. Methods Oper. Res..

[155]  Chen Li,et al.  Feasible region contraction interior point algorithm , 2006, Appl. Math. Comput..

[156]  Nimrod Megiddo,et al.  Introduction: New approaches to linear programming , 1986, Algorithmica.

[157]  Richard S.H. Mah,et al.  5 – SPARSE MATRIX COMPUTATION , 1990 .

[158]  T. Coleman,et al.  The null space problem II. Algorithms , 1987 .

[159]  Wei Li,et al.  A basis-deficiency-allowing primal phase-I algorithm using the most-obtuse-angle column rule , 2006, Comput. Math. Appl..

[160]  Å. Björck,et al.  Large scale matrix problems , 1983 .

[161]  H. Markowitz.  The Elimination form of the Inverse and its Application to Linear Programming , 1957 .

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

[163]  Evan L. Porteus,et al.  A class of methods for linear programming , 1978, Math. Program..

[164]  Pan,et al.  PRIMAL PERTURBATION SIMPLEX ALGORITHMS FOR LINEAR PROGRAMMING , 2000 .

[165]  Robert E. Bixby,et al.  Very Large-Scale Linear Programming: A Case Study in Combining Interior Point and Simplex Methods , 1992, Oper. Res..

[166]  E. Beale.  Cycling in the dual simplex algorithm , 1955 .

[167]  Peter Zörnig,et al.  Systematic construction of examples for cycling in the simplex method , 2006, Comput. Oper. Res..

[168]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[169]  Karl-Heinz Borgwardt,et al.  The Average number of pivot steps required by the Simplex-Method is polynomial , 1982, Z. Oper. Research.

[170]  Irvin Lustig,et al.  Feasibility issues in a primal-dual interior-point method for linear programming , 1990, Math. Program..

[171]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

[172]  Karl Heinz Borgwardt,et al.  Some Distribution-Independent Results About the Asymptotic Order of the Average Number of Pivot Steps of the Simplex Method , 1982, Math. Oper. Res..

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

[174]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[175]  G. Nemhauser,et al.  Integer Programming , 2020 .

[176]  Jerzy Neyman,et al.  Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability : held at the Statistical Laboratory, University of California, December, 1954, July and August, 1955 , 1958 .

[177]  Donald Goldfarb,et al.  Steepest-edge simplex algorithms for linear programming , 1992, Math. Program..

[178]  Higher Order Iteration Schemes for Unconstrained Optimization , 2011 .

[179]  R. Fourer.  Sparse Gaussian Elimination of Staircase Linear Systems. , 1979 .

[180]  L. V. Kantorovich,et al.  Mathematical Methods of Organizing and Planning Production , 1960 .

[181]  Pingqi Pan,et al.  Practical finite pivoting rules for the simplex method , 1990 .

[182]  M. Padberg.  Linear Optimization and Extensions , 1995 .

[183]  Erling D. Andersen,et al.  Presolving in linear programming , 1995, Math. Program..

[184]  Marcos Pereira Estellita Lins,et al.  An improved intial basis for the simplex algorithm , 2005 .

[185]  Shuzhong Zhang,et al.  Pivot rules for linear programming: A survey on recent theoretical developments , 1993, Ann. Oper. Res..

[186]  Shinji Mizuno,et al.  A primal—dual infeasible-interior-point algorithm for linear programming , 1993, Math. Program..

[187]  G. Dantzig,et al.  Notes on Linear Programming: Part 1. The Generalized Simplex Method for Minimizing a Linear Form under Linear Inequality Restraints , 1954 .

[188]  Shinji Mizuno,et al.  An O(√nL)-Iteration Homogeneous and Self-Dual Linear Programming Algorithm , 1994, Math. Oper. Res..

[189]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[190]  N. Andrei,et al.  Balance constraints reduction of large-scale linear programming problems , 1993, Ann. Oper. Res..

[191]  Ping-Qi Pan.  A Revised Dual Projective Pivot Algorithm for Linear Programming , 2005, SIAM J. Optim..

[192]  William W. Hager,et al.  The Dual Active Set Algorithm and Its Application to Linear Programming , 2002, Comput. Optim. Appl..

[193]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[194]  John L. Nazareth,et al.  The implementation of linear programming algorithms based on homotopies , 1996, Algorithmica.

[195]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[196]  Ping-Qi Pan,et al.  A projective simplex algorithm using LU decomposition , 2000 .

[197]  L. S. Srinath.  Linear Programming: Principles and Applications , 1983 .

[198]  E. Balas.  An Additive Algorithm for Solving Linear Programs with Zero-One Variables , 1965 .

[199]  Paula M. J. Harris.  Pivot selection methods of the Devex LP code , 1973, Math. Program..

[200]  Stephen Smale,et al.  On the average number of steps of the simplex method of linear programming , 1983, Math. Program..

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

[202]  T. M. Williams,et al.  Practical Methods of Optimization. Vol. 1: Unconstrained Optimization , 1980 .

[203]  D. Bartholomew,et al.  Linear Programming: Methods and Applications , 1970 .

[204]  George B. Dantzig,et al.  Linear Programming 1: Introduction , 1997 .

[205]  Pingqi Pan.  A simplex-like method with bisection for linear programming 1 , 1991 .

[206]  D. R. Fulkerson,et al.  An algorithm for scaling matrices. , 1962 .

[207]  Tamás Terlaky,et al.  Target-Following Methods for Linear Programming , 1996 .

[208]  Nimrod Megiddo,et al.  A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension , 1985, JACM.

[209]  B. Jansen,et al.  The theory of linear programming:skew symmetric self-dual problems and the central path * , 1994 .

[210]  Jean-Philippe Vial,et al.  A polynomial newton method for linear programming , 2005, Algorithmica.

[211]  A. Tucker,et al.  Linear Inequalities And Related Systems , 1956 .

[212]  Y. Luke,et al.  Padé approximations to the operator exponential , 1970 .

[213]  Shinji Mizuno,et al.  On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming , 1993, Math. Oper. Res..

[214]  Pablo Guerrero-García,et al.  Phase I cycling under the most-obtuse-angle pivot rule , 2005, Eur. J. Oper. Res..

[215]  Ping-Qi Pan,et al.  A phase-1 approach for the generalized simplex algorithm , 2001 .

[216]  István Maros,et al.  Advances in design and implementation of optimization software , 2002, Eur. J. Oper. Res..

[217]  Peter Abel,et al.  On the choice of the pivot columns of the Simplex-method: Gradient criteria , 1987, Computing.

[218]  Ping-Qi Pan,et al.  A Dual Projective Pivot Algorithm for Linear Programming , 2004, Comput. Optim. Appl..

[219]  Yinyu Ye,et al.  Interior point algorithms: theory and analysis , 1997 .

[220]  Robert J. Vanderbei,et al.  A modification of karmarkar's linear programming algorithm , 1986, Algorithmica.

[221]  Mauricio G. C. Resende,et al.  An implementation of Karmarkar's algorithm for linear programming , 1989, Math. Program..

[222]  Philip Wolfe,et al.  Recent Advances in Mathematical Programming , 2011 .

[223]  Michael A. Saunders,et al.  A practical anti-cycling procedure for linearly constrained optimization , 1989, Math. Program..

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

[225]  Hanif D. Sherali,et al.  Linear Programming and Network Flows , 1977 .

[226]  Steve Smale,et al.  The Problem of the Average Speed of the Simplex Method , 1982, ISMP.

[227]  Earl R. Barnes,et al.  A variation on Karmarkar’s algorithm for solving linear programming problems , 1986, Math. Program..

[228]  Yinyu Ye,et al.  A simplified homogeneous and self-dual linear programming algorithm and its implementation , 1996, Ann. Oper. Res..

[229]  L. Liao,et al.  ON AN EFFICIENT IMPLEMENTATION OF THE FACE ALGORITHM FOR LINEAR PROGRAMMING , 2013 .

[230]  J. A. Tomlin,et al.  On scaling linear programming problems , 1975 .

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

[232]  Harvey J. Greenberg,et al.  Design and implementation of optimization software , 1977, Math. Program..

[233]  Jacek Gondzio,et al.  Implementation of Interior Point Methods for Large Scale Linear Programming , 1996 .