Criss-Cross Simplex Method

Methods perform very differently when solving a same problem. It is common that a problem that is solved slowly by the simplex method would be solved fast by the dual simplex method; and vise versa. Consequently, LP packages often include multiple options to be chosen, as it seems to be impossible to predetermine which method would be better to solve a given problem. Pursuing a method with features of both the primal and dual methods, scholars have attempted to combine primal and dual simplex methods for years. A class of resulting variants may be described by “criss-cross” because of their switching between primal and dual iterations. The primal-dual algorithm (Sect. 7.1) and the self-dual parametric simplex algorithm (Sect. 7.2) may be also classified into this category.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[47]  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.

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

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

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

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

[52]  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..

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[79]  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..

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[131]  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..

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

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

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

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

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

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

[138]  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 .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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