Interior point methods for linear optimization

Interior Point Methods for Linear Optimization

Volumn , Issue , 2005, Pages 1-497

  Roos, Cornelis ^a   Terlaky, Tamás ^b   Vial, Jean Philippe ^c  

^a DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

^b MCMASTER UNIVERSITY   (Canada)

^c UNIVERSITY OF GENEVA   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84892218719     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/b100325     Document Type: Book

References (323)

1. I. Adler, N.K. Karmarkar, M.G.C. Resende, and G. Veiga. Data structures and programming techniques for the implementation of Karmarkar's algorithm. ORSA J. on Computing, 1:84-106, 1989.
2. I. Adler, N.K. Karmarkar, M.G.C. Resende, and G. Veiga. An implementation of Karmarkar's algorithm for linear programming. Mathematical Programming, 44:297-335, 1989. Errata in Mathematical Programming, 50:415, 1991.
3. I. Adler and R.D.C. Monteiro. Limiting behavior of the affine scaling continuous trajectories for linear programming problems. Mathematical Programming, 50:29-51, 1991.
4. I. Adler and R.D.C. Monteiro. A geometric view of parametric linear programming. Algorithmica, 8:161-176, 1992.
5. A.V. Aho, J.E. Hopcroft, and J.D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Mass., 1974.
6. M. Akgiil. A note on shadow prices in linear programming. J. Operational Research Society, 35:425-431, 1984.
7. A. Altman and K.C. Kiwiel. A note on some analytic center cutting plane methods for convex feasibility and minimization problems. Computational Optimization and Applications, 5, 1996.
8. E.D. Andersen. Finding all linearly dependent rows in large-scale linear programming. Optimization Methods and Software, 6:219-227, 1995.
9. E.D. Andersen and K.D. Andersen. Presolving in linear programming. Mathematical Programming, 71:221-245, 1995.
10. E.D. Andersen, J. Gondzio, Cs. Meszaros, and X. Xu. Implementation of interior point methods for large scale linear programming. In T. Terlaky, editor. Interior Point Methods of Mathematical Programming, pp. 189-252. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
11. E.D. Andersen and Y. Ye. Combining interior-point and pivoting algorithms for linear programming. Management Science, 42(12): 1719-1731, 1996.
12. K.D. Andersen. A modified Schur complement method for handling dense columns in interior-point methods for linear programming. ACM Transactions Mathematical Software, 22(3):348-356, 1996.
13. E.D. Andersen, C. Roos, T. Terlaky, T. Trafalisand J.P. Warners. The use of low-rank updates in interior-point methods. In: Ed. Y. Yuan, Numerical Linear Algebra and Optimization, pp. 1-12. Science Press, Beijing, China, 2004.
14. K.M. Anstreicher. A monotonic projective algorithm for fractional linear programming. Algorithmica, 1(4):483-498, 1986.
15. K.M. Anstreicher. A strengthened acceptance criterion for approximate projections in Karmarkar's algorithm. Operations Research Letters, 5:211-214, 1986.
16. K.M. Anstreicher. A combined Phase I - Phase II projective algorithm for linear programming. Mathematical Programming, 43:209-223, 1989.
17. K.M. Anstreicher. Progress in interior point algorithms since 1984. SI AM News, 22:12- 14, March 1989.
18. K.M. Anstreicher. The worst-case step in Karmarkar's algorithm. Mathematics of Operations Research, 14:294-302, 1989.
19. K.M. Anstreicher. Dual ellipsoids and degeneracy in the projective algorithm for linear programming. Contemporary Mathematics, 114:141-149, 1990.
20. K.M. Anstreicher. A standard form variant and safeguarded linesearch for the modified Karmarkar algorithm. Mathematical Programming, 47:337-351, 1990.
21. K.M. Anstreicher. A combined phase I - phase II scaled potential algorithm for linear programming. Mathematical Programming, 52:429-439, 1991.
22. K.M. Anstreicher. On the performance of Karmarkar's algorithm over a sequence of iterations. SIAM J. on Optimization, 1(1):22-29, 1991.
23. K.M. Anstreicher. On interior algorithms for linear programming with no regularity assumptions. Operations Research Letters, 11:209-212, 1992.
24. K.M. Anstreicher. Potential reduction algorithms. In T. Terlaky, editor. Interior Point Methods of Mathematical Programming, pp. 125-158. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
25. K.M. Anstreicher and R.A. Bosch. Long steps in a 0{n3L) algorithm for linear programming. Mathematical Programming, 54:251-265, 1992.
26. K.M. Anstreicher and R.A. Bosch. A new infinity-norm path following algorithm for linear programming. SIAM J. on Optimization, 5:236-246, 1995.
27. M. Arioli, I.S. Duff, and P.P.M. de Rijk. On the augmented system approach to sparse least-squares problems. Numer. Math., 55:667-684, 1989.
28. M.D. Asic, V.V. Kovacevic-Vujcic, and M.D. Radosavljevic-Nikolic. A note on limiting behavior of the projective and the affine rescaling algorithms. Contemporary Mathematics, 114:151-157, 1990.
29. D.S. Atkinson and P.M. Vaidya. A scaling technique for finding the weighted analytic center of a polytope. Mathematical Programming, 57:163-192, 1992.
30. D.S. Atkinson and P.M. Vaidya. A cutting plane algorithm that uses analytic centers. Mathematical Programming, 69(69), 1995.
31. O. Bahn, J.-L. Goffin, O. du Merle, and J.-Ph. Vial. A cutting plane method from analytic centers for stochastic programming. Mathematical Programming, 69(1):45-73, 1995.
32. Y.Q. Bai, M. Elghami, and C. Roos. A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization. SIAM J. on Optimization, 15(1):101-128, 2004.
33. M.L. Balinski and A.W. Tucker. Duality theory of linear programs: a constructive approach with applications. SIAM Review, 11:499-581, 1969.
34. E.R. Barnes. A variation on Karmarkar's algorithm for solving linear programming problems. Mathematical Programming, 36:174-182, 1986.
35. E.R. Barnes. Some results concerning convergence of the affine scaling algorithm. Contemporary Mathematics, 114:131-139, 1990.
36. E.R. Barnes, S. Chopra, and D.J. Jensen. The affine scaling method with centering. Technical Report, Dept. of Mathematical Sciences, IBM T. J. Watson Research Center, P. O. Box 218, Yorktown Heights, NY 10598, USA, 1988.
37. M.S. Bazaraa, H.D. Sherali, and CM. Shetty. Nonlinear Programming: Theory and Algorithms. John Wiley & Sons, New York (second edition), 1993.
38. R. Bellman. Introduction to Matrix Analysis. Volume 12 of SIAM Classics in Applied Mathematics. SIAM, Philadelphia, 1995.
39. A. Ben-Israel and T.N.E. Greville. Generalized Inverses: Theory and Applications. John Wiley & Sons, New York, USA, 1974.
40. D.P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, Massachusetts, 1995.
41. G. Birkhoff and S. MacLane. A Survey of Modern Algebra. Macmillan, New York, 1977.
42. R.E. Bixby. Progress in linear programming. ORSA J. on Computing, 6:15-22, 1994.
43. R.E. Bixby and M.J. Saltzman. Recovering an optimal LP basis from an interior point solution. Operations Research Letters, 15(4): 169-178, 1994.
44. A. Bjork. Numerical Methods for Least Squares Problems. SIAM, Philadelphia, 1996.
45. J.F. Bonnans and F.A. Potra. Infeasible path-following algorithms for linear complementarity problems. Mathematics of Operations Research, 22(2), 378-407, 1997.
46. R.A. Bosch. On Mizuno's rank one updating algorithm for linear programming. SIAM J. on Optimization, 3:861-867, 1993.
47. R.A. Bosch and K.M. Anstreicher. On partial updating in a potential reduction linear programming algorithm of Kojima, Mizuno and Yoshise. Algorithmica, 9(1): 184-197, 1993.
48. S.E. Boyd and L. Vandenberghe. Semidefinite programming. SIAM Review, 38(1):49- 96, 1996.
49. A.L. Brearley, G. Mitra, and H.P. Williams. Analysis of mathmetical programming problems prior to applying the simplex algorithm. Mathematical Programming, 15:54-83, 1975.
50. M.G. Breitfeld and D.F. Shanno. Computational experiene with modified log-barrier methods for nonlinear programming. Annals of Operations Research, 62:439-464, 1996.
51. J. Brinkhuis and G. Draisma. Schiet Op, Special Report, Econometric Institute, Erasmus University, Rotterdam, 1996.
52. R.C. Buck. Advanced Calculus. International Series in Pure and Apllied Mathematics. Mac-Graw Hih Book Company, New York (third edition), 1978.
53. J.R. Bunch and B.N. Parlett. Direct methods for solving symmetric indefinit systems of linear equations. SIAM J. on Numerical Analysis, 8:639-655, 1971.
54. S.F. Chang and S.T. McCormick. A hierachical algorithm for making sparse matrices sparse. Mathematicaal Programming, 56:1-30, 1992.
55. V. Chvatal. Linear programming. W.H. Freeman and Company, New York, USA, 1983.
56. S.A. Cook. The complexity of theorem-proving procedures. In Proceedings of Third Annual ACM Symposium on Theory of Computing, pp. 151-158. ACM, New York, 1971.
57. J.P. Crouzeix and C. Roos. On the inverse target map of a linear programming problem. Unpublished Manuscript, University of Clermont, France, 1994.
58. I. Csiszar. /-divergence geometry of probability distributions and minimization problems. Annals of Probability, 3:146-158, 1975.
59. G.B. Dantzig. Linear Programming and Extensions. Princeton Univ. Press, Princeton, New Jersey, 1963.
60. G.B. Dantzig. Linear programming. In J.K. Lenstra, A.H.G. Rinnooy Kan, and A. Schrijver, editors. History of mathmetical programming. A collection of personal reminiscences. CWI, North-Holland, The Netherlands, 1991.
61. A. Deza, E. Nematollahi, R. Peyghami and T. Terlaky. The central path visits all the vertices of the Klee-Minty cube. AdvOl-Report #2004/11. McMaster Univ., Hamilton, Ontario, Canada.
62. A. Deza, E. Nematollahi and T. Terlaky. How good are interior point methods? Klee- Minty cubes tighten iteration-complexity bounds. AdvOl-Report #2004/20. McMaster Univ., Hamilton, Ontario, Canada.
63. I.I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747-748, 1967. Translated in Soviet Mathematics Doklady, 8:674-675, 1967.
64. I.I. Dikin. On the convergence of an iterative process. Upravlyaemye Sistemi, 12:54-60, 1974. (In Russian).
65. I.I. Dikin. Letter to the editor. Mathematical Programming, 41:393-394, 1988.
66. LI. Dikin and C. Roos. Convergence of the dual varaibles for the primal affine scaling method with unit steps in the homogeneous case. J. of Optimization Theory and Applications, 95:305-321, 1997.
67. J. Ding and T.Y. Li. An algorithm based on weighted logarithmic barrier functions for linear complementarity problems. Arabian J. for Science and Engineering, 15(4) :679- 685, 1990.
68. I.S. Duff. The solution of large-scale least-squares problems on supercomputers. Annals of Operations Research, 22:241-252, 1990.
69. I.S. Duff, N.I.M. Gould, J.K. Reid, J.A. Scott, and K. Turner. The factorization of sparse symmetric indefinite matrices. IMA J. on Numerical Analysis, 11:181-204, 1991.
70. A.S. El-Bakry, R.A. Tapia, and Y. Zhang. A study of indicators for identifying zero variables in interior-point methods. SI AM Review, 36(1):45-72, 1994.
71. A.S. El-Bakry, R.A. Tapia, and Y. Zhang. On the convergence rate of Newton interiorpoint methods in the absence of strict complementarity. Computational optimization and Applications, 6:157-167, 1996.
72. J.R. Evans and N.R. Baker. Degeneracy and the (mis)interpretation of sensitivity analysis in linear programming. Decision Sciences, 13:348-354, 1982.
73. J.R. Evans and N.R. Baker. Reply to 'On ranging cost coefficients in dual degenerate linear programming problems'. Decision Sciences, 14:442-443, 1983.
74. S.-Ch. Fang and S. Puthenpura. Linear optimization and extensions: theory and algorithms. Prentice Hall, Englewood Cliffs, New Jersey 07632, 1993.
75. J. Farkas. Theorie der einfachen Ungleichungen. J. Reine und Angewandte Mathematik, 124:1-27, 1902.
76. A.V. Fiacco. Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, volume 165 of Mathematics in Science and Engineering. Academic Press, New York, 1983.
77. A.V. Fiacco and G.P. McCormick. Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley & Sons, New York, 1968. Reprint: Volume 4 of SIAM Classics in Applied Mathematics, SIAM Publications, Philadelphia, PA 19104- 2688, USA, 1990.
78. R. Fourer and S. Mehrotra. Solving symmetric indefinite systems in an interior-point method for linear programming. Mathematical Programming, 62:15-40, 1993.
79. C. Fraley. Linear updates for a single-phase projective method. Operations Research Letters, 9:169-174, 1990.
80. C. Fraley and J.-Ph. Vial. Numerical study of projective methods for linear programming. In S. Dolecki, editor. Optimization: Proceedings of the 5th French-German Conference in Castel-Novel, Varetz, France, October 1988, volume 1405 of Lecture Notes in Mathematics, pp. 25-38. Springer Verlag, Berlin, West-Germany, 1989.
81. C. Fraley and J.-Ph. Vial. Alternative approaches to feasibility in projective methods for linear programming. ORSA J. on Computing, 4:285-299, 1992.
82. P. Franklin. A Treatise on Advanced Calculus. John Wiley & Sons, New York (fifth edition), 1955.
83. R.M. Freund. An analogous of Karmarkar's algorithm for inequality constrained linear programs, with a 'new' class of projective transformations for centering a polytope. Operations Research Letters, 7:9-13, 1988.
84. R.M. Freund. Theoretical efficiency of a shifted barrier function algorithm for linear programming. Linear Algebra and Its Applications, 152:19-41, 1991.
85. R.M. Freund. Projective transformation for interior-point algorithms, and a superlinearly convergent algorithm for the w-center problem. Mathematical Programming, 58:385-414, 1993.
86. K.R. Frisch. Principles of linear programming - the double gradient form of the logarithmic potential method. Memorandum, Institute of Economics, University of Oslo, Oslo, Norway, October 1954.
87. K.R. Frisch. La resolution des problemes de programme lineaire par la methode du potential logarithmique. Cahiers du Seminaire D,Econometric, 4:7-20, 1956.
88. K.R. Frisch. The logarithmic potential method for solving linear programming problems. Memorandum, University Institute of Economics, Oslo, 1955.
89. T. Gal. Postoptimal analyses, parametric programming and related topics. Mac-Graw Hill Inc., New York/Berlin, 1979.
90. T. Gal. Shadow prices and sensitivity analysis in linear programming under degeneracy, state-of-the-art-survey. OR Spektrum, 8:59-71, 1986.
91. D. Gale. The Theory of Linear Economic Models. McGraw-Hill, New York, USA, 1960.
92. M.R. Garey and D.S. Johnson. Computers and Intractability: a Guide to the Theory of NP-completeness. Freeman, San Francisco, 1979.
93. J. Gauvin. Quelques precisions sur les prix marginaux en programmation lineaire. INFOR, 18:68-73, 1980. (In French).
94. A. George and J.W.-H. Liu. Computing Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs, NJ, 1981.
95. G. de Ghellinck and J.-Ph. Vial. A polynomial Newton method for linear programming. Algorithmica, 1(4):425-453, 1986.
96. G. de Ghellinck and J.-Ph. Vial. An extension of Karmarkar's algorithm for solving a system of linear homogeneous equations on the simplex. Mathematical Programming, 39:79-92, 1987.
97. P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin, and M.H. Wright. On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method. Mathematical Programming, 36:183-209, 1986.
98. J.-L. Goffin, J. Gondzio, R. Sarkissian, and J.-Ph. Vial. Solving nonlinear multicommodity flow problems by the analytic center cutting plane method. Mathematical Programming, 76:131-154, 1997.
99. J.-L. Goffin, A. Haurie, and J.-Ph. Vial. Decomposition and nondifferentiable optimization with the projective algorithm. Management Science, 38(2):284-302, 1992.
100. J.-L. Goffin, Z.-Q. Luo, and Y. Ye. Complexity analysis of an interior cutting plane for convex feasibility problems. SIAM J. on Optimization, 6(3), 1996.
101. J.-L. Goffin and F. Sharifi-Mokhtarian-Mokhtarian. Primal-dual-infeasible Newton approach for the analytic center deep-cutting plane method. J. Optim. Theory AppL, 101(1):35-58, 1999.
102. J.-L. Goffin and J.-Ph. Vial. On the computation of weighted analytic centers and dual ellipsoids with the projective algorithm. Mathematical Programming, 60:81-92, 1993.
103. J.-L. Goffin and J.-Ph. Vial. Short steps with Karmarkar's projective algorithm for linear programming. SIAM J. on Optimization, 4:193-207, 1994.
104. J.-L. Goffin and J.-Ph. Vial. Shallow, deep and very deep cuts in the analytic center cutting plane method. Math. Program., 84(1, Ser. A):89-103, 1999.
105. D. Goldfarb and S. Mehrotra. Relaxed variants of Karmarkar's algorithm for linear programs with unknown optimal objective value. Mathematical Programming, 40:183- 195, 1988.
106. D. Goldfarb and S. Mehrotra. A relaxed version of Karmarkar's method. Mathematical Programming, 40:289-315, 1988.
107. D. Goldfarb and S. Mehrotra. A self-correcting version of Karmarkar's algorithm. SIAM J. on Numerical Analysis, 26:1006-1015, 1989.
108. D. Goldfarb and D.X. Shaw. On the complexity of a class of projective interior point methods. Mathematics of Operations Research, 20:116-134, 1995.
109. D. Goldfarb and M.J. Todd. Linear Programming. In G.L. Nemhauser, A.H.G. Rinnooy Kan, and M.J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science, pp. 141-170. North Holland, Amsterdam, The Netherlands, 1989.
110. D. Goldfarb and D. Xiao. A primal projective interior point method for linear programming. Mathematical Programming, 51:17-43, 1991.
111. A.J. Goldman and A.W. Tucker. Theory of linear programming. In H.W. Kuhn and A.W. Tucker, editors. Linear Inequalities and Related Systems, Annals of Mathematical Studies, No. 38, pp. 53-97. Princeton University Press, Princeton, New Jersey, 1956.
112. G.H. Golub and C.F. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore (second edition), 1989.
113. J. Gondzio. Presolve analysis of linear programs prior to applying the interior point method. INFORMS J. on Computing, 9:73-91,1997.
114. J. Gondzio. Multiple centrality corrections in a primal-dual method for linear programming. Computational Optimization and Applications, 6:137-156, 1996.
115. J. Gondzio, O. du Merle, R. Sarkissian, and J.-Ph. Vial. ACCPM - a library for convex optimization based on an analytic center cutting plane method. European J. of Operational Research, 94:206-211, 1996.
116. J. Gondzio and T. Terlaky. A computational view of interior point methods for linear programming. In J.E. Beasley, editor. Advances in Linear and Integer Programming, pp. 103-185. Oxford University Press, Oxford, Great Britain, 1996.
117. C.C. Gonzaga. A simple representation of Karmarkar's algorithm. Technical Report, Dept. of Systems Engineering and Computer Science, COPPE Federal University of Rio de Janeiro, 21941 Rio de Janeiro, RJ, Brazil, May 1988.
118. C.C. Gonzaga. An algorithm for solving linear programming problems in 0(n3L) operations. In N. Megiddo, editor. Progress in Mathematical Programming: Interior Point and Related Methods, pp. 1-28. Springer Verlag, New York, 1989.
119. C.C. Gonzaga. Conical projection algorithms for linear programming. Mathematical Programming, 43:151-173, 1989.
120. C.C. Gonzaga. Convergence of the large step primal affine-scaling algorithm for primal nondegenerate linear programs. Technical Report ES-230/90, Dept. of Systems Engineering and Computer Science, COPPE Federal University of Rio de Janeiro, 21941 Rio de Janeiro, RJ, Brazil, September 1990.
121. C.C. Gonzaga. Large step path-following methods for linear programming. Part I: Barrier function method. SI AM J. on Optimization, 1:268-279, 1991.
122. C.C. Gonzaga. Large step path-following methods for linear programming. Part II: Potential reduction method. SIAM J. on Optimization, 1:280-292, 1991.
123. C.C. Gonzaga. Search directions for interior linear programming methods. Algorithmica, 6:153-181, 1991.
124. C.C. Gonzaga. Path-following methods for linear programming. SIAM Review, 34(2): 167-227, 1992.
125. C.C. Gonzaga. The largest step path following algorithm for monotone linear complementarity problems. Mathematical Programming, 76(2):309-332, 1997.
126. C.C. Gonzaga and R.A. Tapia. On the convergence of the Mizuno-Todd-Ye algorithm to the analytic center of the solution set. SIAM J. on Optimization, 7: 47-65, 1997.
127. C.C. Gonzaga and R.A. Tapia. On the quadratic convergence of the simplified Mizuno- Todd-Ye algorithm for linear programming. SIAM J. on Optimization, 7:66-85, 1997.
128. H.J. Greenberg. An analysis of degeneracy. Naval Research Logistics Quarterly, 33:635- 655, 1986.
129. H.J. Greenberg. The use of the optimal partition in a linear programming solution for postoptimal analysis. Operations Research Letters, 15:179-186, 1994.
130. O. Giiler, 1994. Private communication.
131. O. Giiler. Limiting behavior of the weighted central paths in linear programming. Mathematical Programming, 65(2):347-363, 1994.
132. O. Giiler, D. den Hertog, C. Roos, T. Terlaky, and T. Tsuchiya. Degeneracy in interior point methods for linear programming: A survey. Annals of Operations Research, 46:107-138, 1993.
133. O. Giiler, C. Roos, T. Terlaky, and J.-Ph. Vial. Interior point approach to the theory of linear programming. Cahiers de Recherche 1992.3, Faculte des Sciences Economique et Sociales, Universite de Geneve, Geneve, Switzerland, 1992.
134. O. Giiler, C. Roos, T. Terlaky, and J.-Ph. Vial. A survey of the implications of the behavior of the central path for the duality theory of linear programming. Management Science, 41:1922-1934, 1995.
135. O. Giiler and Y. Ye. Convergence behavior of interior-point algorithms. Mathematical Programming, 60(2):215-228, 1993.
136. W. W. Eager. Updating the inverse of a matrix. SIAM Review, 31(2):221-239, June 1989.
137. M. Halicka. Analytical properties of the central path at the boundary point in linear programming. Mathematical Programming, 84:335-355, 1999.
138. L.A. Hall and R.J. Vanderbei. Two-third is sharp for affine scaling. Operations Research Letters, 13:197-201, 1993.
139. G.H. Hardy, J.E. Littlewood, and G. Polya. Inequalities. Cambridge University Press, Cambridge, Cambridge, 1934.
140. D. den Hertog. Interior Point Approach to Linear, Quadratic and Convex Programming, volume 277 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
141. D. den Hertog, J.A. Kaliski, C. Roos, and T. Terlaky. A logarithmic barrier cutting plane method for convex programming. Annals of Operations Reasearch, 58:69-98, 1995.
142. D. den Hertog and C. Roos. A survey of search directions in interior point methods for linear programming. Mathematical Programming, 52:481-509, 1991.
143. D. den Hertog, C. Roos, and T. Terlaky. A potential reduction variant of Renegar's short-step path-following method for linear programming. Linear Algebra and Its Applications, 152:43-68, 1991.
144. D. den Hertog, C. Roos, and T. Terlaky. On the monotonicity of the dual objective along barrier paths. COAL Bulletin, 20:2-8, 1992.
145. D. den Hertog, C. Roos, and T. Terlaky. Adding and deleting constraints in the logarithmic barrier method for LP. In D.-Z. Du and J. Sun, editors. Advances in Optimization and Approximation, pp. 166-185. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
146. D. den Hertog, C. Roos, and J.-Ph. Vial. A complexity reduction for the long-step path-following algorithm for linear programming. SIAM J. on Optimization, 2:71-87, 1992.
147. R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, UK, 1985.
148. P. Huard. Resolution of mathmetical programming with nonlinear constraints by the method of centers. In J. Abadie, editor. Nonlinear Programming, pp. 207-219. North Holland, Amsterdam, The Netherlands, 1967.
149. P. Huard. A method of centers by upper-bounding functions with applications. In J.B. Rosen, O.L. Mangasarian, and K. Ritter, editors. Nonlinear Programming: Proceedings of a Symposium held at the University of Wisconsin, Madison, Wisconsin, USA, May 1970, pp. 1-30. Academic Press, New York, USA, 1970.
150. P. Hung and Y. Ye. An asymptotically (9(y,L)-iteration path-following linear programming algorithm that uses long steps. SIAM J. on Optimization, 6:570-586, 1996.
151. B. Jansen. Interior Point Techniques in Optimization. Complexity, Sensitivity and Algorithms volume 6 of Applied Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
152. B. Jansen, J.J. de Jong, C. Roos, and T. Terlaky. Sensitivity analysis in linear programming: just be careful! European Journal of Operations Research 101:15-28, 1997.
153. B. Jansen, C. Roos, and T. Terlaky. An interior point approach to postoptimal and parametric analysis in linear programming. Technical Report 92-21, Faculty of Technical Mathematics and Computer Science, TU Delft, NL-2628 CD Delft, The Netherlands, April 1992.
154. B. Jansen, C. Roos, and T. Terlaky. A family of polynomial affine scaling algorithms for positive semi-definite linear complementarity problems. SIAM J. on Optimization, 7(1):126-140, 1997.
155. B. Jansen, C. Roos, and T. Terlaky. The theory of linear programming : Skew symmetric self-dual problems and the central path. Optimization, 29:225-233, 1994.
156. B. Jansen, C. Roos, and T. Terlaky. A polynomial Dikin-type primal-dual algorithm for linear programming. Mathematics of Operations Research, 21:341-353, 1996.
157. B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal-dual algorithms for linear programming based on the logarithmic barrier method. J. of Optimization Theory and Applications, 83:1-26, 1994.
158. B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Long-step primal-dual target-following algorithms for linear programming. Mathematical Methods of Operations Research, 44:11-30, 1996.
159. B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal-dual target-following algorithms for linear programming. Annals of Operations Research, 62:197-231, 1996.
160. B. Jansen, C. Roos, T. Terlaky, and Y. Ye. Improved complexity using higher-order correctors for primal-dual Dikin affine scaling. Mathematical Programming, 76:117-130, 1997.
161. F. Jarre. Interior-point methods for classes of convex programs. In T. Terlaky, editor. Interior Point Methods of Mathematical Programming, pp. 255-296. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
162. F. Jarre, M. Kocvara, and J. Zowe. Optimal truss design by interior-point methods. SIAM J. on Optimization, 8(4):1084-1107, 1998.
163. F. Jarre and M.A. Saunders. An adaptive primal-dual method for linear programming. COAL Newsletter, 19:7-16, August 1991.
164. J. Kaliski, D. Haglin, C. Roos, and T. Terlaky. Logarithmic barrier decomposition methods for semi-infinite programming. International Transactions in Operations Research 4:285-303, 1997.
165. N.K. Karmarkar. A new polynomial-time algorithm for linear programming. Combinatorica, 4:373-395, 1984.
166. R.M. Karp. Reducibility among combinatorial problems. In R.E. Miller and J.W. Thatcher, editors. Complexity of computer computations, pp. 85-103. Plenum Press, New York, 1972.
167. L.G. Khachiyan. A polynomial algorithm in linear programming. Doklady Akademiia Nauk SSSR, 244:1093-1096, 1979.
168. Translated into English in Soviet Mathematics Doklady 20, 191-194.
169. K.C. Kiwiel. Complexity of some cutting plane methods that use analytic centers. Mathematical Programming, 74(1), 1996.
170. E. Klafszky, J. Mayer, and T. Terlaky. Linearly constrained estimation by mathmetical programming. European J. of Operational Research, 34:254-267, 1989.
171. E. Klafszky and T. Terlaky. On the ellipsoid method. Szigma, 20(2-3): 196-208, 1988. In Hungarian.
172. E. Klafszky and T. Terlaky. The role of pivoting in proving some fundamental theorems of linear algebra. Linear Algebra and its Applications, 151:97-118, 1991.
173. E. de Klerk, C. Roos, and T. Terlaky. A nonconvex weighted potential function for polynomial target following methods. Annals of Operations Reasearch, 81:3-14, 1998.
174. G. Knolmayer. The effects of degeneracy on cost-coefficient ranges and an algorithm to resolve interpretation problems. Decision Sciences, 15:14-21, 1984.
175. M. Kojima, N. Megiddo, and S. Mizuno. A primal-dual infeasible-interior- point algorithm for linear programming. Mathematical Programming, 61:263-280, 1993.
176. M. Kojima, N. Megiddo, T. Noma, and A. Yoshise. A unified approach to interior point algorithms for linear complementarity problems, volume 538 of Lecture Notes in Computer Science. Springer Verlag, Berlin, Germany, 1991.
177. M. Kojima, S. Mizuno, and T. Noma. Limiting behavior of trajectories by a continuation method for monotone complementarity problems. Mathematics of Operations Research, 15(4):662-675, 1990.
178. M. Kojima, S. Mizuno, and A. Yoshise. A polynomial-time algorithm for a class of linear complementarity problems. Mathematical Programming, 44:1-26, 1989.
179. M. Kojima, S. Mizuno, and A. Yoshise. A primal-dual interior point algorithm for linear programming. In N. Megiddo, editor. Progress in Mathematical Programming: Interior Point and Related Methods, pp. 29-47. Springer Verlag, New York, 1989.
180. E. Kranich. Interior point methods for mathmetical programming: A bibliography. Discussion Paper 171, Institute of Economy and Operations Research, FernUniversitat Hagen, P.O. Box 940, D-5800 Hagen 1, West-Germany, May 1991. Available through NETLIB, see Kranich [180].
181. E. Kranich. Interior-point methods bibliography. SIAG/OPT Views-and-News, A Forum for the SI AM Activity Group on Optimization, 1:11, 1992.
182. P. Lancester and M. Tismenetsky. The Theory of Matrices with Applications. Academic Press, Orlando (second edition), 1985.
183. P.D. Ling. A new proof of convergence for the new primal-dual affine scaling interiorpoint algorithm of Jansen, Roos and Terlaky. Working paper. University of East-Anglia, Norwich, England, 1993.
184. P.D. Ling. A predictor-corrector algorithm. Working Paper, University of East Anglia, Norwich, England, 1993.
185. C.L. Liu. Introduction to Combinatorial Mathematics. Mac-Graw Hill Book Company, New York, 1968.
186. F.A. Lootsma. Numerical Methods for Nonlinear Optimization. Academic Press, London, UK, 1972.
187. Z.-Q. Luo. Analysis of a cutting plane method that uses weighted analytic center an multiple cuts. SI AM J. on Optimization, 7(3):697-716, 1997.
188. Z.-Q. Luo, C. Roos, and T. Terlaky. Complexity analysis of a logarithmic barrier decomposition method for semi-infinite linear programming. Applied Numerical Mathematics, 29:379-394, 1999.
189. Z.-Q. Luo and Y. Ye. A genuine quadratically convergent polynomial interior point algorithm for linear programming. In D.-Z. Du and J. Sun, editors. Advances in Optimization and Approximation, pp. 235-246. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
190. LJ. Lustig. An analysis of an available set of linear programming test problems. Computers and Operations Research, 16:173-184, 1989.
191. LJ. Lustig. Phase 1 search directions for a primal-dual interior point method for linear programming. Contemporary Mathematics, 114:121-130, 1990.
192. LJ. Lustig, R.E. Marsten, and D.F. Shanno. Computational experience with a primaldual interior point method for linear programming. Linear Algebra and Its Applications, 152:191-222, 1991.
193. LJ. Lustig, R.E. Marsten, and D.F. Shanno. On implementing Mehrotra's predictorcorrector interior point method for linear programming. SIAM J. on Optimization, 2:435-449, 1992.
194. LJ. Lustig, R.E. Marsten, and D.F. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA J. on Computing, 6(1):1-14, 1994.
195. H.M. Markowitz. The elimination form of the inverse and its application to linear programming. Management Science, 3:255-269, 1957.
196. I. Maros and Cs. Meszaros. The role of the augmented system in interior point methods. European J. of Operational Research, 107(3):720-736, 1998.
197. R.E. Marsten, D.F. Shanno and E.M. Simantiraki. Interior point methods for linear and nonlinear programming. In LA. Duff and A. Watson, editors. The state of the art in numerical analysis (York, 1996), volume 63 of Inst. Math. Appl. Conf. Ser. New Ser., pages 339-362. Oxford Univ. Press, New York, 1997.
198. L. McLinden. The analogue of Moreau's proximation theorem, with applications to the nonlinear complementarity problem. Pacific J. of Mathematics, 88:101-161, 1980.
199. L. McLinden. The complementarity problem for maximal monotone multifunctions. In R.W. Cottle, F. Giannessi, and J.L. Lions, editors. Variational Inequalities and Complementarity Problems, pp. 251-270. John Wiley and Sons, New York, 1980.
200. K.A. McShane, C.L. Monma, and D.F. Shanno. An implementation of a primal-dual interior point method for linear programming. ORSA J. on Computing, 1:70-83, 1989.
201. N. Megiddo. Pathways to the optimal set in linear programming. In N. Megiddo, editor. Progress in Mathematical Programming: Interior Point and Related Methods, pp. 131-158. Springer Verlag, New York, 1989.
202. Identical version in Proceedings of the 6th Mathematical Programming Symposium of Japan, Nagoya, Japan, pp. 1-35, 1986.
203. N. Megiddo. On finding primal- and dual-optimal bases. ORSA J. on Computing, 3:63-65, 1991.
204. S. Mehrotra. Higher order methods and their performance. Technical Report 90-16R1, Dept. of Industrial Engineering and Management Science, Northwestern University, Evanston, IL 60208, USA, 1990. Revised July 1991.
205. S. Mehrotra. Finding a vertex solution using an interior point method. Linear Algebra and Its Applications, 152:233-253, 1991.
206. S. Mehrotra. Deferred rank-one updates in 0{n,L) interior point algorithm. J. of the Operations Research Society of Japan, 35:345-352, 1992.
207. S. Mehrotra. On the implementation of a (primal-dual) interior point method. SI AM J. on Optimization, 2(4):575-601, 1992.
208. S. Mehrotra. Quadratic convergence in a primal-dual method. Mathematics of Operations Research, 18:741-751, 1993.
209. S. Mehrotra and R.D.C. Monteiro. Parametric and range analysis for interior point methods. Mathematical Programming, 74:65-82, 1996.
210. S. Mehrotra and Y. Ye. On finding the optimal facet of linear programs. Mathematical Programming, 62:497-515, 1993.
211. O. du Merle. Mise en oeuvre et developpements de la methode de plans coupants bases sur les centres analytiques. PhD thesis, Faculte des Sciences Economiques et Sociales, Universite de Geneve, 1995. In French.
212. H.D. Mills. Marginal values of matrix games and linear programs. In H.W. Kuhn and A.W. Tucker, editors. Linear Inequalities and Related Systems, Annals of Mathematical Studies, No. 38, pp. 183-193. Princeton University Press, Princeton, New Jersey, 1956.
213. J.E. Mitchell. Fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems. European J. of Operational Research, 97:139-148, 1997.
214. S. Mizuno. An 0(n,L) algorithm using a sequence for linear complementarity problems. J. of the Operations Research Society of Japan, 33:66-75, 1990.
215. S. Mizuno. A rank-one updating interior algorithm for linear programming. Arabian J. for Science and Engineering, 15(4):671-677, 1990.
216. S. Mizuno. A new polynomial time method for a linear complementarity problem. Mathematical Programming, 56:31-43, 1992.
217. S. Mizuno. A primal-dual interior point method for linear programming. The Proceeding of the Institute of Statistical Mathematics, 40(1):27-44, 1992. In Japanese.
218. S. Mizuno and M.J. Todd. An 0(n,L) adaptive path following algorithm for a linear complementarity problem. Mathematical Programming, 52:587-595, 1991.
219. S. Mizuno, M.J. Todd, and Y. Ye. On adaptive-step primal-dual interior-point algorithms for linear programming. Mathematics of Operations Research, 18:964-981, 1993.
220. R.D.C. Monteiro and I. Adler. Interior-path following primal-dual algorithms: Part I: Linear programming. Mathematical Programming, 44:27-41, 1989.
221. R.D.C. Monteiro and I. Adler. Interior path-following primal-dual algorithms: Part II: Convex quadratic programming. Mathematical Programming, 44:43-66, 1989.
222. R.D.C. Monteiro, I. Adler, and M.G.C. Resende. A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension. Mathematics of Operations Research, 15:191-214, 1990.
223. R.D.C. Monteiro and S. Mehrotra. A general parametric analysis approach and its implications to sensitivity analysis in interior point methods. Mathematical Programming, 72:65-82, 1996.
224. R.D.C. Monteiro and T. Tsuchiya. Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem. Mathematics of Operations Research 21(4):793-814, 1996.
225. M. Muramatsu and T. Tsuchiya. Convergence analysis of the projective scaling algorithm based on a long-step homogeneous affine scaling algorithm. Mathematical Programming, 72:291-305, 1996.
226. G.L. Nemhauser and L.A. Wolsey. Integer and Combinatorial Optimization. J. Wiley & Sons, New York, 1988.
227. Y. Nesterov. Cutting plane algorithms from analytic centers: efficiency estimates. Mathematical Programming, 69(1), 1995.
228. Nesterov and A.S. Nemirovskii. Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms. SIAM Publications. SIAM, Philadelphia, USA, 1993.
229. J. von Neumann. On a maximization problem. Manuscript, Institute for Advanced Studies, Princeton University, Princeton, NJ 08544, USA, 1947.
230. F. Nozicka, J. Guddat, H. Hollatz, and B. Bank. Theorie der linearen parametrischen Optimierung. Akademie-Verlag, Berlin, 1974.
231. M.R. Osborne. Finite Algorithms in Optimization and Data Analysis. John Wiley & Sons, New York, USA, 1985.
232. M. Padberg. Linear Optimization and Extensions, volume 12 of Algorithmis and Combinatorics. Springer Verlag, Berlin, West-Germany, 1995.
233. C.H. Papadimitriou and K. Steiglitz. Combinatorial optimization. Algorithms and complexity. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1982.
234. J. Peng. Private communication.
235. Peng, C. Roos and T. Terlaky. Self-Regularity: A New Paradigm for Primal-Dual Interior Point Methods. Princeton University Press, 2002.
236. R. Polyak. Modified barrier functions (theory and methods). Mathematical Programming, 54:177-222, 1992.
237. F.A. Potra. A quadratically convergent predictor-corrector method for solving linear programs from infeasible starting points. Mathematical Programming, 67(3):383-406, 1994.
238. M.V. Ramana and P.M. Pardalos. Semidefinite programming. In T. Terlaky, editor. Interior Point Methods of Mathematical Programming, pp. 369-398. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
239. J. Renegar. A polynomial-time algorithm, based on Newton's method, for linear programming. Mathematical Programming, 40:59-93, 1988.
240. R.T. Rockafellar. The elementary vectors of a subspace of IR, In R.C. Bose and T.A. Bowling, editors. Combinatorial Mathematics and Its Applications: Proceedings North Caroline Conference, Chapel Hill, 1967, pp. 104-127. The University of North Caroline Press, Chapel Hill, North Carohne, 1969.
241. C. Roos. New trajectory-following polynomial-time algorithm for linear programming problems. J. of Optimization Theory and Applications, 63:433-458, 1989.
242. C. Roos. An 0(n,L) approximate center method for linear programming. In S. Dolecki, editor. Optimization: Proceedings of the 5th French-German Conference in Castel- Novel, Varetz, France, October 1988, volume 1405 of Lecture Notes in Mathematics, pp. 147-158. Springer Verlag, Berlin, West-Germany, 1989.
243. C. Roos and D. den Hertog. A polynomial method of approximate weighted centers for linear programming. Technical Report 89-13, Faculty of Mathematics and Computer Science, TU Delft, NL-2628 BL Delft, The Netherlands, 1989.
244. C. Roos and T. Terlaky. Advances in linear optimization. In M. Dell'Amico, F. Maffioli, and S. Martello, editors. Annotated Bibliographies in Combinatorial Optimization, pp. 95-114. John Wiley & Sons, New York, USA, 1997.
245. C. Roos and J.-Ph. Vial. Analytic centers in linear programming. Technical Report 88-74, Faculty of Mathematics and Computer Science, TU Delft, NL-2628 BL Delft, The Netherlands, 1988.
246. C. Roos and J.-Ph. Vial. Long steps with the logarithmic penalty barrier function in linear programming. In J. Gabszevwicz, J.F. Richard, and L. Wolsey, editors. Economic Decision-Making: Games, Economics and Optimization, dedicated to J.H. Dreze, pp. 433-441. Elsevier Science Publisher B.V., Amsterdam, The Netherlands, 1989.
247. C. Roos and J.-Ph. Vial. A polynomial method of approximate centers for linear programming. Mathematical Programming, 54:295-305, 1992.
248. C. Roos and J.-Ph. Vial. Achievable potential reductions in the method of Kojima et al. in the case of linear programming. Revue RAIRO-Operations Research, 28:123-133, 1994.
249. D.S. Rubin and H.M. Wagner. Shadow prices: tips and traps for managers and instructors. Interfaces, 20:150-157, 1990.
250. W. Rudin. Principles of Mathematical Analysis. Mac-Craw Hill Book Company, New York, 1978.
251. R. Saigal. Linear Programming, A modern integrated analysis. International series in operations research & management. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995.
252. A. Schrijver. Theory of Linear and Integer Programming. John Wiley & Sons, New York, 1986.
253. D.F. Shanno. Computing Karmarkar projections quickly. Mathematical Programming, 41:61-71, 1988.
254. D.F. Shanno, M.C. Breitfeld, and E.M. Simantiraki. Implementing barrier methods for nonlinear programming. In T. Terlaky, editor. Interior Point Methods of Mathematical Programming, pp. 369-398. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
255. R. Sharda. Linear programming software for personal computers: 1995 survey. OR/MS Today, pp. 49-57, October 1995.
256. D.X. Shaw and D. Coldfarb. A path-following projective interior point method for linear programming. SIAM J. on Optimization, 4:65-85, 1994.
257. N.Z. Shor. Quadratic optimization problems. Soviet J. of Computer and System Sciences, 25:1-11, 1987.
258. C. Sierksma. Linear and integer programming, volume 245 oi Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York, second edition, 2002. Theory and practice. With 1 IBM-PC floppy disk ("INTPM, a version of Karmarkar's Interior Point Method") by J. Cjaltema and C. A. Tijssen (3.5 inch; HD).
259. Gy. Sonnevend. An "analytic center" for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In A. Prekopa, J. Szelezsan, and B. Strazicky, editors, System Modelling and Optimization: Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985, volume 84 of Lecture Notes in Control and Information Sciences, pp. 866-876. Springer Verlag, Berlin, West- Germany, 1986.
260. Gy. Sonnevend, J. Stoer, and G. Zhao. On the complexity of following the central path by linear extrapolation in linear programming. Methods of Operations Research, 62:19-31, 1990.
261. G. Strang. Linear Algebra and its Applications. Harcourt Brace Jovanovich, Orlando, Florida, USA, 1988.
262. J.F. Sturm and S. Zhang. An 0(,/nL) iteration bound primal-dual cone affine scaling algorithm. Mathematical Programming, 72:177-194, 1996.
263. K. Tanabe. Centered Newton methods and Differential Geometry of Optimization. Cooperative Research Report 89. The Institute of Statistical Mathematics, Tokyo, Japan, 1996. (Contains 38 papers related to the subject).
264. M.J. Todd. Recent developments and new directions in linear programming. In M. Iri and K. Tanabe, editors. Mathematical Programming: Recent Developments and Applications, pp. 109-157. Kluwer Academic Press, Dordrecht, The Netherlands, 1989.
265. M.J. Todd. The effects of degeneracy, null and unbounded variables on variants of Karmarkar's linear programming algorithm. In T.F. Coleman and Y. Li, editors, Large- Scale Numerical Optimization. Volume 46 of SI AM Proceedings in Applied Mathematics, pp. 81-91. SIAM, Philadelphia, PA, USA, 1990.
266. M.J. Todd. A lower bound on the number of iterations of primal-dual interior-point methods for linear programming. In G.A. Watson and D.F. Griffiths, editors. Numerical Analysis 1993, volume 303 of Pitman Research Notes in Mathematics, pp. 237-259. Longman Press, Harlow, 1994. See also [267].
267. M.J. Todd. Potential-reduction methods in mathmetical programming. Mathematical Programming, 76 (1), 3-45, 1997.
268. M.J. Todd and B.P. Burrell. An extension of Karmarkar's algorithm for linear programming using dual variables. Algorithmica, 1(4):409-424, 1986.
269. M.J. Todd and Y. Ye. A lower bound on the number of iterations of long-step and polynomial interior-point linear programming algorithms. Annals of Operations Research, 62:233-252, 1996.
270. T. Tsuchiya. Global convergence of the affine scaling methods for degenerate linear programming problems. Mathematical Programming, 52:377-404, 1991
271. T. Tsuchiya. Degenerate linear programming problems and the affine scaling method. Systems, Control and Information, 34(4):216-222, April 1990. (In Japanese).
272. T. Tsuchiya. Global convergence property of the affine scaling methods for primal degenerate linear programming problems. Mathematics of Operations Research, 17(3) :527- 557, 1992.
273. T. Tsuchiya. Quadratic convergence of Iri-Imai algorithm for degenerate linear programming problems. J. of Optimization Theory and Applications, 87(3):703-726, 1995.
274. T. Tsuchiya. Affine scaling algorithm. In T. Terlaky, editor. Interior Point Methods of Mathematical Programming, pp. 35-82. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
275. T. Tsuchiya and M. Muramatsu. Global convergence of the long-step affine scaling algorithm for degenerate linear programming problems. SIAM J. on Optimization, 5(3):525-551, 1995.
276. A.W. Tucker. Dual systems of homogeneous linear relations. In H.W. Kuhn and A.W. Tucker, editors, Linear Inequalities and Related Systems, Annals of Mathematical Studies, No. 38, pp. 3-18. Princeton University Press, Princeton, New Jersey, 1956.
277. K. Turner. Computing projections for the Karmarkar algorithm. Linear Algebra and Its Applications, 152:141-154, 1991.
278. P.M. Vaidya. An algorithm for linear programming which requires 0((m + n)n2{m + n1.5',nL) arithmetic operations. Mathematical Programming, 47:175-201, 1990.
279. R.J. Vanderbei. Symmetric quasi-definite matrices. SIAM J. on Optimization, 5(1): 100-113, 1995.
280. R.J. Vanderbei and T.J. Carpenter. Symmetric indefinite systems for interior point methods. Mathematical Programming, 58:1-32, 1993.
281. R.J. Vanderbei, M.S. Meketon, and B.A. Freedman. A modification of Karmarkar's linear programming algorithm. Algorithmica, 1(4):395-407, 1986.
282. S.A. Vavasis and Y. Ye. Condition numbers for polyhedra with real number data. Operations Research Letters, 17:209-214, 1995.
283. S.A. Vavasis and Y. Ye. A primal-dual interior point method whose running time depends only on the constraint matrix. Mathematical Programming, 74:79-120, 1996.
284. J.-Ph. Vial. A fully polynomial time projective method. Operations Research Letters, 7(1), 1988.
285. J.-Ph. Vial. A unified approach to projective algorithms for linear programming. In S. Dolecki, editor. Optimization: Proceedings of the 5th French-German Conference in Castel-Novel, Varetz, France, October 1988, volume 1405 of Lecture Notes in Mathematics, pp. 191-220. Springer Verlag, Berlin, West-Germany, 1989.
286. J.-Ph. Vial. A projective algorithm for linear programming with no regularity condition. Operations Research Letters, 12(1), 1992.
287. J.-Ph. Vial. A generic path-following algorithm with a sliding constraint and its application to linear programming and the computation of analytic centers. Technical Report 1996.8, LOGILAB/Management Studies, University of Geneva, Switzerland, 1996.
288. J.-Ph. Vial. A path-following version of the Todd-Burrell procedure for linear programming. Mathematical Methods of Operations Research, 46(2): 153-167, 1997.
289. G.R. Walsh. An Introduction to Linear Programming. John Wiley & Sons, New York, USA, 1985.
290. J.E. Ward and R.E. Wendell. Approaches to sensitivity analysis in linear programming. Annals of Operations Research, 27:3-38, 1990.
291. D.S. Watkins. Fundamentals of Matrix Computations. John Wiley & Sons, New York, 1991.
292. M. Wechs. The analyticity of interior-point-paths at strictly complementary solutions of linear programs. Optimization Methods and Software, 9:209-243, 1998.
293. A.C. Williams. Boundedness relations for linear constraints sets. Linear Algebra and Its Applications, 3:129-141, 1970.
294. A.C. Williams. Complenientarity theorems for linear programming. SI AM Review, 12:135-137, 1970.
295. H.P. Williams. Model Building in Mathematical Programming. John Wiley & Sons, New York, USA (third edition), 1990.
296. C. Witzgall, P.T. Boggs, and P.D. Domich. On the convergence behavior of trajectories for linear programming. Contemporary Mathematics, 114:161-187, 1990.
297. S.J. Wright. An infeasible-interior-point algorithm for linear complementarity problems. Mathematical Programming, 67(1):29-52, 1994.
298. S.J. Wright and D. Ralph. A superlinear infeasible-interior-point algorithm for monotone nonlinear complementarity problems. Mathematics of Operations Research, 21(4):815-838, 1996.
299. S.J. Wright. A path-following infeasible-interior-point algorithm for linear complementarity problems. Optimization Methods and Software, 2:79-106, 1993.
300. S.J. Wright. Primal-Dual Interior-Point Methods. SIAM, Philadelphia, 1996.
301. F. Wu, S. Wu, and Y. Ye. On quadratic convergence of the O(v,L)-iteration homogeneous and self-dual linear programming algorithm. Annals of Operations Research, 87: 393-406, 1999.
302. S.R. Xu, H.B. Yao, and Y.Q. Chen. An improved Karmarkar algorithm for linear programming and its numerical tests. Mathematica Applicata, 5(1): 14-21, 1992. (In Chinese, English summary).
303. H. Yamashita. A polynomially and quadratically convergent method for linear programming. Working Paper, Mathematical Systems Institute, Inc., Tokyo, Japan, 1986.
304. M. Yannakakis. Computing the minimum fill-in is NP-complete. SIAM J. on Algebraic Discrete Methods, pp. 77-79, 1981.
305. Y. Ye. Interior algorithms for linear, quadratic, and linearly constrained convex programming. PhD thesis, Dept. of Engineering Economic Systems, Stanford University, Stanford, CA 94305, USA, 1987.
306. Y. Ye. Karmarkar's algorithm and the ellipsoid method. Operations Research Letters, 6:177-182, 1987.
307. Y. Ye. A class of projective transformations for linear programming. SIAM J. on Computing, 19:457-466, 1990.
308. Y. Ye. An 0(n,L) potential reduction algorithm for linear programming. Mathematical Programming, 50:239-258, 1991.
309. Y. Ye. Extensions of the potential reduction algorithm for linear programming. J. of Optimization Theory and Applications, 72(3):487-498, 1992.
310. Y. Ye. On the finite convergence of interior-point algorithms for linear programming. Mathematical Programming, 57:325-335, 1992.
311. Y. Ye. On the g'-order of convergence of interior-point algorithms for linear programming. In Wu Fang, editor. Proceedings Symposium on Applied Mathematics. Chinese Academy of Sciences, Institute of Applied Mathematics, 1992.
312. Y. Ye. A potential reduction algorithm allowing column generation. SIAM J. on Optimization, 2:7-20, 1992.
313. Y. Ye. Toward probabilistic analysis of interior-point algorithms for linear programming. Mathematics of Operations Research, 19:38-52, 1994.
314. Y. Ye. Complexity analysis of the analytic center cutting plane method that uses multiple cuts. Mathematical Programming, 76(1):211-221, 1997.
315. Y. Ye, O. Giiler, R.A. Tapia, and Y. Zhang. A quadratically convergent 0{√n,L)- iteration algorithm for linear programming. Mathematical Programming, 59:151-162, 1993.
316. Y. Ye and P.M. Pardalos. A class of linear complementarity problems solvable in polynomial time. Linear Algebra and Its Applications, 152:3-17, 1991.
317. Y. Ye and M.J. Todd. Containing and shrinking ellipsoids in the path-following algorithm. Mathematical Programming, 47:1-10, 1990.
318. Y. Ye, M.J. Todd, and S. Mizuno. An O(y,L)-iteration homogeneous and self-dual linear programming algorithm. Mathematics of Operations Research, 19:53-67, 1994.
319. Y. Ye, O. Giiler, R.A. Tapia, and Y. Zhang. A quadratically convergent 0{√nL) iteration algorithm for linear programming. Mathematical Programming, 59:151-162, 1993.
320. L. Zhang and Y. Zhang. On polynomiality of the Mehrotra-type predictor-corrector interior-point algorithms. Mathematical Programming, 68:303-318, 1995.
321. Y. Zhang and R.A. Tapia. Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited. J. of Optimization Theory and Applications, 73(2):229-242, 1992.
322. G. Zhao. Interior point algorithms for linear complementarity problems based on large neighborhoods of the central path. SIAM J. on Optimization, 8(2), 397-413, 1998.
323. G. Zhao and J. Zhu. Analytical properties of the central trajectory in interior point methods. In D-Z. Du and J. Sun, editors. Advances in Optimization and Approximation, pp. 362-375. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.