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Linear and Nonlinear Programming pp 11–31Cite as

Basic Properties of Linear Programs

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 228))

Abstract

A linear program (LP) is an optimization problem in which the objective function is linear in the unknowns and the constraints consist of linear equalities and linear inequalities. The exact form of these constraints may differ from one problem to another, but as shown below, any linear program can be transformed into the following standard form:

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Notes

  1. 1.

    See Appendix A for a description of the vector notation used throughout this book.

Bibliography

  1. J. Abadie, J. Carpentier, Generalization of the Wolfe reduced gradient method to the case of nonlinear constraints, in Optimization, ed. by R. Fletcher (Academic, London, 1969), pp. 37–47

    Google Scholar 

  2. H. Akaike, On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method. Ann. Inst. Stat. Math. 11, 1–17 (1959)

    Google Scholar 

  3. A.Y. Alfakih, A. Khandani, H. Wolkowicz, Solving euclidean distance matrix completion problems via semidefinite programming. Comput. Opt. Appl. 12, 13–30 (1999)

    Google Scholar 

  4. F. Alizadeh, Combinatorial optimization with interior point methods and semi-definite matrices, Ph.D. thesis, University of Minnesota, Minneapolis, 1991

    Google Scholar 

  5. F. Alizadeh, Optimization over the positive semi-definite cone: interior-point methods and combinatorial applications, in Advances in Optimization and Parallel Computing, ed. by P.M. Pardalos (North Holland, Amsterdam, 1992), pp. 1–25

    Google Scholar 

  6. E.D. Andersen, MOSEK: high performance software for large-scale LP, QP, SOCP, SDP and MIP, http://www.mosek.com/ (1997)

  7. E.D. Andersen, Y. Ye, On a homogeneous algorithm for the monotone complementarity problem. Math. Prog. 84, 375–400 (1999)

    Google Scholar 

  8. K.M. Anstreicher, D. den Hertog, C. Roos, T. Terlaky, A long step barrier method for convex quadratic programming. Algorithmica 10, 365–382 (1993)

    Google Scholar 

  9. H.A. Antosiewicz, W.C. Rheinboldt, Numerical analysis and functional analysis, in Survey of Numerical Analysis, ed. by J. Todd, Chap. 14 (McGraw-Hill, New York, 1962)

    Google Scholar 

  10. L. Armijo, Minimization of functions having lipschitz continuous first-partial derivatives. Pac. J. Math. 16(1), 1–3 (1966)

    Google Scholar 

  11. K.J. Arrow, L. Hurwicz, Gradient method for concave programming, I.: local results, in Studies in Linear and Nonlinear Programming, ed. by K.J. Arrow, L. Hurwicz, H. Uzawa (Stanford University Press, Stanford, CA, 1958)

    Google Scholar 

  12. E. Balas, S. Ceria, G. Cornuejols, A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Program. 58, 295–324 (1993)

    Google Scholar 

  13. R.H. Bartels, A numerical investigation of the simplex method. Technical Report No. CS 104, Computer Science Department, Stanford University, Stanford, CA (31 July 1968)

    Google Scholar 

  14. R.H. Bartels, G.H Golub, The simplex method of linear programming using LU decomposition. Commun. ACM 12(5), 266–268 (1969)

    Google Scholar 

  15. A. Barvinok, A remark on the rank of positive semidefinite matrices subject to affine constraints. Discrete Comput. Geom. 25, 23–31 (2001)

    Google Scholar 

  16. A. Barvinok, A Course in Convexity. Graduate Studies in Mathematics, vol. 54 (American Mathematical Society, Providence, RI, 2002)

    Google Scholar 

  17. J. Barzilai, J.M. Borwein, Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (2008)

    Google Scholar 

  18. D.A., Bayer, J.C. Lagarias, The nonlinear geometry of linear programming, part i: affine and projective scaling trajectories. Trans. Am. Math. Soc 314(2), 499–526 (1989)

    Google Scholar 

  19. D.A. Bayer, J.C. Lagarias, The nonlinear geometry of linear programming, part ii: legendre transform coordinates. Trans. Am. Math. Soc. 314(2), 527–581 (1989)

    Google Scholar 

  20. M.S. Bazaraa, J.J. Jarvis, Linear Programming and Network Flows (Wiley, New York, 1977)

    Google Scholar 

  21. M.S. Bazaraa, J.J. Jarvis, H.F. Sherali, Karmarkar’s projective algorithm (Chap. 8.4), pp. 380–394; Analysis of Karmarkar’s algorithm (Chap. 8.5), pp. 394–418, in Linear Programming and Network Flows, 2nd edn. (Wiley New York, 1990)

    Google Scholar 

  22. E.M.L. Beale, in Numerical Methods, Nonlinear Programming, ed. by J. Abadie (North-Holland, Amsterdam, 1967)

    Google Scholar 

  23. A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)

    Google Scholar 

  24. F.S. Beckman, The solution of linear equations by the conjugate gradient method, in Mathematical Methods for Digital Computers, ed. by A. Ralston, H.S. Wilf, vol. 1 (Wiley, New York, 1960)

    Google Scholar 

  25. S.J. Benson, Y. Ye, X. Zhang, Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10, 443–461 (2000)

    Google Scholar 

  26. A. Ben-Tal, A. Nemirovski, Structural design via semidefinite programming, in Handbook on Semidefinite Programming (Kluwer, Boston, 2000), pp. 443–467

    Google Scholar 

  27. D.P. Bertsekas, Partial conjugate gradient methods for a class of optimal control problems. IEEE Trans. Autom. Control 19, 209–217 (1973)

    Google Scholar 

  28. D.P. Bertsekas, Multiplier methods: a survey. Automatica 12(2), 133–145 (1976)

    Google Scholar 

  29. D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic, New York, 1982)

    Google Scholar 

  30. D.P. Bertsekas, Nonlinear Programming (Athena Scientific, Belmont, 1995)

    Google Scholar 

  31. D. Bertsimas, Y. Ye, Semidefinite relaxations, multivariate normal distributions, and order statistics, in Handbook of Combinatorial Optimization (Springer, New York, 1999), pp. 1473–1491

    Google Scholar 

  32. D.M. Bertsimas, J.N. Tsitsiklis, Linear Optimization (Athena Scientific, Belmont, 1997)

    Google Scholar 

  33. M.C. Biggs, Constrained minimization using recursive quadratic programming: some alternative sub-problem formulations, in Towards Global Optimization, ed. by L.C.W. Dixon, G.P. Szego (North-Holland, Amsterdam, 1975)

    Google Scholar 

  34. M.C. Biggs, On the convergence of some constrained minimization algorithms based on recursive quadratic programming. J. Inst. Math. Appl. 21, 67–81 (1978)

    Google Scholar 

  35. G. Birkhoff, Three observations on linear algebra. Rev. Univ. Nac. Tucumán, Ser. A. 5, 147–151 (1946)

    Google Scholar 

  36. P. Biswas, Y. Ye, Semidefinite programming for ad hoc wireless sensor network localization, in Proceedings of the 3rd IPSN, 2004, pp. 46–54

    Google Scholar 

  37. R.E. Bixby, Progress in linear programming. ORSA J. Comput. 6(1), 15–22 (1994)

    Google Scholar 

  38. R.G. Bland, New finite pivoting rules for the simplex method. Math. Oper. Res. 2(2), 103–107 (1977)

    Google Scholar 

  39. R.G. Bland, D. Goldfarb, M.J. Todd, The ellipsoidal method: a survey. Oper. Res. 29, 1039–1091 (1981)

    Google Scholar 

  40. L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation (Springer, New York, 1996)

    Google Scholar 

  41. S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3, 1–122 (2010)

    Google Scholar 

  42. S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Science (SIAM, Philadelphia, 1994)

    Google Scholar 

  43. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  44. C.G. Broyden, Quasi-Newton methods and their application to function minimization. Math. Comput. 21, 368–381 (1967)

    Google Scholar 

  45. C.G. Broyden, The convergence of a class of double rank minimization algorithms: parts I and II. J. Inst. Math. Appl. 6, 76–90, 222–231 (1970)

    Google Scholar 

  46. T. Butler, A.V. Martin, On a method of courant for minimizing functionals. J. Math. Phys. 41, 291–299 (1962)

    Google Scholar 

  47. C.W. Carroll, The created response surface technique for optimizing nonlinear restrained systems. Oper. Res. 9(12), 169–184 (1961)

    Google Scholar 

  48. A. Charnes, Optimality and degeneracy in linear programming. Econometrica 20, 160–170 (1952)

    Google Scholar 

  49. A. Charnes, C.E. Lemke, The bounded variables problem. ONR Research Memorandum 10, Graduate School of Industrial Administration, Carnegie Institute of Technology, Pittsburgh (1954)

    Google Scholar 

  50. C.H. Chen, B.S. He, Y.Y. Ye, X.M. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math. Program. (2014). doi:10.1007/s10107-014-0826-5

    Google Scholar 

  51. A. Cohen, Rate of convergence for root finding and optimization algorithms. Ph.D. dissertation, University of California, Berkeley, 1970

    Google Scholar 

  52. S.A. Cook, The complexity of theorem-proving procedures, in Proceedings of 3rd ACM Symposium on the Theory of Computing, 1971, pp. 151–158

    Google Scholar 

  53. R.W. Cottle, Linear Programming. Lecture Notes for MS& E 310 (Stanford University, Stanford, 2002)

    Google Scholar 

  54. R. Cottle, J.S. Pang, R.E. Stone, Interior-Point Methods (Chap. 5.9), in The Linear Complementarity Problem (Academic, Boston, 1992), pp. 461–475

    Google Scholar 

  55. R. Courant, Calculus of variations and supplementary notes and exercises (mimeographed notes), supplementary notes by M. Kruskal and H. Rubin, revised and amended by J. Moser, New York University (1962)

    Google Scholar 

  56. J.B. Crockett, H. Chernoff, Gradient methods of maximization. Pac. J. Math. 5, 33–50 (1955)

    Google Scholar 

  57. H. Curry, The method of steepest descent for nonlinear minimization problems. Q. Appl. Math. 2, 258–261 (1944)

    Google Scholar 

  58. Y.-H. Dai, R. Fletcher, Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. Numer. Math. 100, 21–47 (2005)

    Google Scholar 

  59. J.W. Daniel, The conjugate gradient method for linear and nonlinear operator equations. SIAM J. Numer. Anal. 4(1), 10–26 (1967)

    Google Scholar 

  60. G.B. Dantzig, Maximization of a linear function of variables subject to linear inequalities (Chap. XXI), in Activity Analysis of Production and Allocation, ed. by T.C. Koopmans. Cowles Commission Monograph, vol. 13 (Wiley, New York, 1951)

    Google Scholar 

  61. G.B., Dantzig, Application of the simplex method to a transportation problem, in Activity Analysis of Production and Allocation, ed. by T.C. Koopmans (Wiley, New York, 1951), pp. 359–373

    Google Scholar 

  62. G.B. Dantzig, Computational algorithm of the revised simplex method. RAND Report RM-1266, The RAND Corporation, Santa Monica, CA (1953)

    Google Scholar 

  63. G.B. Dantzig, Variables with upper bounds in linear programming. RAND Report RM-1271, The RAND Corporation, Santa Monica, CA (1954)

    Google Scholar 

  64. G.B. Dantzig, Linear Programming and Extensions (Princeton University Press, Princeton, 1963)

    Google Scholar 

  65. G.B. Dantzig, L.R. Ford Jr., D.R. Fulkerson, A primal-dual algorithm, in Linear Inequalities and Related Systems. Annals of Mathematics Study, vol. 38 (Princeton University Press, Princeton, NJ, 1956), pp. 171–181

    Google Scholar 

  66. G.B. Dantzig, A. Orden, P. Wolfe, Generalized simplex method for minimizing a linear form under linear inequality restraints. RAND Report RM-1264, The RAND Corporation, Santa Monica, CA (1954)

    Google Scholar 

  67. G.B. Dantzig, M.N. Thapa, Linear Programming 1: Introduction (Springer, New York, 1997)

    Google Scholar 

  68. G.B. Dantzig, M.N. Thapa, Linear Programming 2: Theory and Extensions (Springer, New York, 2003)

    Google Scholar 

  69. G.B. Dantzig, P. Wolfe, Decomposition principle for linear programs. Oper. Res. 8, 101–111 (1960)

    Google Scholar 

  70. W.C. Davidon, Variable metric method for minimization. Research and Development Report ANL-5990 (Ref.) U.S. Atomic Energy Commission, Argonne National Laboratories (1959)

    Google Scholar 

  71. W.C. Davidon, Variance algorithm for minimization. Comput. J. 10, 406–410 (1968)

    Google Scholar 

  72. E. de Klerk, C. Roos, T. Terlaky, Initialization in semidefinite programming via a self–dual skew–symmetric embedding. Oper. Res. Lett. 20, 213–221 (1997)

    Google Scholar 

  73. R.S. Dembo, S.C. Eisenstat, T. Steinhaug, Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982)

    Google Scholar 

  74. J.E. Dennis Jr., J.J. Moré, Quasi-Newton methods, motivation and theory. SIAM Rev. 19, 46–89 (1977)

    Google Scholar 

  75. J.E. Dennis Jr., R.E. Schnabel, Least change secant updates for quasi-Newton methods. SIAM Rev. 21, 443–469 (1979)

    Google Scholar 

  76. L.C.W. Dixon, Quasi-Newton algorithms generate identical points. Math. Prog. 2, 383–387 (1972)

    Google Scholar 

  77. B.C. Eaves, W.I. Zangwill, Generalized cutting plane algorithms. Working Paper No. 274, Center for Research in Management Science, University of California, Berkeley (July 1969)

    Google Scholar 

  78. J. Eckstein, D.P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)

    Google Scholar 

  79. H. Everett III, Generalized lagrange multiplier method for solving problems of optimum allocation of resources. Oper. Res. 11, 399–417 (1963)

    Google Scholar 

  80. D.K. Faddeev, V.N. Faddeeva, Computational Methods of Linear Algebra (W. H. Freeman, San Francisco, CA, 1963)

    Google Scholar 

  81. S.C. Fang, S. Puthenpura, Linear Optimization and Extensions (Prentice-Hall, Englewood Cliffs, NJ, 1994)

    Google Scholar 

  82. W. Fenchel, Convex Cones, Sets, and Functions. Lecture Notes (Department of Mathematics, Princeton University, Princeton, NJ, 1953)

    Google Scholar 

  83. A.V. Fiacco, G.P. McCormick, Nonlinear Programming: Sequential Uncon-strained Minimization Techniques (Wiley, New York, 1968)

    Google Scholar 

  84. A.V. Fiacco, G.P. McCormick, Nonlinear Programming: Sequential Uncon-strained Minimization Techniques (Wiley, New York, 1968). Reprint: Volume 4 of SIAM Classics in Applied Mathematics (SIAM Publications, Philadelphia, PA, 1990)

    Google Scholar 

  85. R. Fletcher, A new approach to variable metric algorithms. Comput. J. 13(13), 317–322 (1970)

    Google Scholar 

  86. R. Fletcher, An exact penalty function for nonlinear programming with inequalities. Math. Program. 5, 129–150 (1973)

    Google Scholar 

  87. R. Fletcher, Conjugate gradient methods for indefinite systems. Numerical Analysis Report, 11. Department of Mathematics, University of Dundee, Scotland (September 1975)

    Google Scholar 

  88. R. Fletcher, Practical Methods of Optimization 1: Unconstrained Optimization (Wiley, Chichester, 1980)

    Google Scholar 

  89. R. Fletcher, Practical Methods of Optimization 2: Constrained Optimization (Wiley, Chichester, 1981)

    Google Scholar 

  90. R. Fletcher, M.J.D. Powell, A rapidly convergent descent method for minimization. Comput. J. 6, 163–168 (1963)

    Google Scholar 

  91. R. Fletcher, C.M. Reeves, Function minimization by conjugate gradients. Comput. J. 7, 149–154 (1964)

    Google Scholar 

  92. L.K. Ford Jr., D.K. Fulkerson, Flows in Networks (Princeton University Press, Princeton, NJ, 1962)

    Google Scholar 

  93. G.E. Forsythe, On the asymptotic directions of the s-dimensional optimum gradient method. Numer. Math. 11, 57–76 (1968)

    Google Scholar 

  94. G.E. Forsythe, C.B. Moler, Computer Solution of Linear Algebraic Systems (Prentice-Hall, Englewood Cliffs, NJ, 1967)

    Google Scholar 

  95. G.E. Forsythe, W.R. Wasow, Finite-Difference Methods for Partial Differential Equations (Wiley, New York, 1960)

    Google Scholar 

  96. M. Fortin, R. Glowinski, On decomposition-coordination methods using an augmented Lagrangian, in Augmented Lagrangian Methods: Applications to the Solution of Boundary Problems, ed. by M. Fortin, R. Glowinski (North- Holland, Amsterdam, 1983)

    Google Scholar 

  97. K. Fox, An Introduction to Numerical Linear Algebra (Clarendon Press, Oxford, 1964)

    Google Scholar 

  98. M. Frank, P. Wolfe, An algorithm for quadratic programming. Naval Res. Logist. Q. 3, 95–110 (1956)

    Google Scholar 

  99. R.M. Freund, Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function. Math. Program. 51, 203–222 (1991)

    Google Scholar 

  100. K.R. Frisch, The logarithmic potential method for convex programming. Unpublished Manuscript, Institute of Economics, University of Oslo, Oslo (1955)

    Google Scholar 

  101. D. Gabay, Reduced quasi-Newton methods with feasibility improvement for nonlinear constrained optimization, in Mathematical Programming Studies, vol. 16 (North-Holland, Amsterdam, 1982), pp. 18–44

    Google Scholar 

  102. D. Gabay, B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2, 17–40 (1976)

    Google Scholar 

  103. D. Gale, The Theory of Linear Economic Models (McGraw-Hill, New York, 1960)

    Google Scholar 

  104. U.M. Garcia-Palomares, O.L. Mangasarian, Superlinearly convergent quasi-Newton algorithms for nonlinearly constrained optimization problems. Math. Program. 11, 1–13 (1976)

    Google Scholar 

  105. S.I. Gass, Linear Programming, 3rd edn. (McGraw-Hill, New York, 1969)

    Google Scholar 

  106. P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin, M.H. Wright, On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method. Math. Program. 36, 183–209 (1986)

    Google Scholar 

  107. P.E., Gill, W. Murray, Quasi-Newton methods for unconstrained optimization. J. Inst. Math. Appl. 9, 91–108 (1972)

    Google Scholar 

  108. P.E. Gill, W. Murray, M.H. Wright, Practical Optimization (Academic, London, 1981)

    Google Scholar 

  109. R. Glowinski, A. Marrocco, Approximation par éléments finis d’ordre un et résolution par pénalisation-dualité d’une classe de problèmes non linéaires. R.A.I.R.O. R2 2, 41–76 (1975)

    Google Scholar 

  110. M.X. Goemans, D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42, 1115–1145 (1995)

    Google Scholar 

  111. D. Goldfarb, A family of variable metric methods derived by variational means. Math. Comput. 24, 23–26 (1970)

    Google Scholar 

  112. D. Goldfarb, G. Iyengar, Robust portfolio selection problems. Math. Oper. Res. 28, 1–38 (2002)

    Google Scholar 

  113. D. Goldfarb, M.J. Todd, Linear programming, in Optimization, ed. by G.L. Nemhauser, A.H.G. Rinnooy Kan, M.J. Todd. Handbooks in Operations Research and Management Science, vol. 1 (North Holland, Amsterdam, 1989), pp. 141–170

    Google Scholar 

  114. D. Goldfarb, D. Xiao, A primal projective interior point method for linear programming. Math. Program. 51, 17–43 (1991)

    Google Scholar 

  115. A.A. Goldstein, On steepest descent. SIAM J. Control 3, 147–151 (1965)

    Google Scholar 

  116. C.C. Gonzaga, An algorithm for solving linear programming problems in O(n 3 L) operations, in Progress in Mathematical Programming: Interior Point and Related Methods, ed. by N. Megiddo (Springer, New York, 1989), pp. 1–28

    Google Scholar 

  117. C.C. Gonzaga, M.J. Todd, An \(O(\sqrt{n}L)\)-iteration large-step primal-dual affine algorithm for linear programming. SIAM J. Optim. 2, 349–359 (1992)

    Google Scholar 

  118. J. Greenstadt, Variations on variable metric methods. Math. Comput. 24, 1–22 (1970)

    Google Scholar 

  119. G. Hadley, Linear Programming (Addison-Wesley, Reading, MA, 1962)

    Google Scholar 

  120. G. Hadley, Nonlinear and Dynamic Programming (Addison-Wesley, Reading, MA, 1964)

    Google Scholar 

  121. S.P. Han, A globally convergent method for nonlinear programming. J. Optim. Theory Appl. 22(3), 297–309 (1977)

    Google Scholar 

  122. H. Hancock, Theory of Maxima and Minima (Ginn, Boston, 1917)

    Google Scholar 

  123. J. Hartmanis, R.E. Stearns, On the computational complexity of algorithms. Trans. Am. Math. Soc. 117, 285–306 (1965)

    Google Scholar 

  124. B.S. He, X.M. Yuan, On the O(1∕n) convergence rate of the Douglas-Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)

    Google Scholar 

  125. D. den Hertog, Interior point approach to linear, quadratic and convex programming, algorithms and complexity, Ph.D. thesis, Faculty of Mathematics and Informatics, TU Delft, BL Delft, 1992

    Google Scholar 

  126. M.R. Hestenes, The conjugate gradient method for solving linear systems, in Proceeding of Symposium in Applied Mathematics, vol. VI, Numerical Analysis (McGraw-Hill, New York 1956), pp. 83–102

    Google Scholar 

  127. M.R. Hestenes, Multiplier and gradient methods. J. Opt. Theory Appl. 4(5), 303–320 (1969)

    Google Scholar 

  128. M.R. Hestenes, Conjugate-Direction Methods in Optimization (Springer, Berlin, 1980)

    Google Scholar 

  129. M.R. Hestenes, E.L. Stiefel, Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. Sect. B 49, 409–436 (1952)

    Google Scholar 

  130. F.L. Hitchcock, The distribution of a product from several sources to numerous localities. J. Math. Phys. 20, 224–230 (1941)

    Google Scholar 

  131. P. Huard, Resolution of mathematical programming with nonlinear constraints by the method of centers, in Nonlinear Programming, ed. by J. Abadie (North Holland, Amsterdam, 1967), pp. 207–219

    Google Scholar 

  132. H.Y. Huang, Unified approach to quadratically convergent algorithms for function minimization. J. Optim. Theory Appl. 5, 405–423 (1970)

    Google Scholar 

  133. L. Hurwicz, Programming in linear spaces, in Studies in Linear and Nonlinear Programming, ed. by K.J. Arrow, L. Hurwicz, H. Uzawa (Stanford University Press, Stanford, CA, 1958)

    Google Scholar 

  134. E. Isaacson, H.B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966)

    Google Scholar 

  135. W. Jacobs, The caterer problem. Naval Res. Logist. Q. 1, 154–165 (1954)

    Google Scholar 

  136. F. Jarre, Interior-point methods for convex programming. Appl. Math. Optim. 26, 287–311 (1992)

    Google Scholar 

  137. W.B. Johnson, J. Lindenstrauss, Extensions of lipshitz mapping into Hilbert space. Comtemp. Math. 26, 189–206 (1984)

    Google Scholar 

  138. S. Karlin, Mathematical Methods and Theory in Games, Programming, and Economics, vol. I (Addison-Wesley, Reading, MA, 1959)

    Google Scholar 

  139. N.K. Karmarkar, A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)

    Google Scholar 

  140. J.E. Kelley, The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. VIII(4), 703–712 (1960)

    Google Scholar 

  141. L.G. Khachiyan, A polynomial algorithm for linear programming. Doklady Akad. Nauk USSR 244, 1093–1096 (1979). Translated in Soviet Math. Doklady 20, 191–194 (1979)

    Google Scholar 

  142. V. Klee, G.J. Minty, How good is the simplex method, in Inequalities III, ed. by O. Shisha (Academic, New York, 1972)

    Google Scholar 

  143. M. Kojima, S. Mizuno, A. Yoshise, A polynomial-time algorithm for a class of linear complementarity problems. Math. Program. 44, 1–26 (1989)

    Google Scholar 

  144. M. Kojima, S. Mizuno, A. Yoshise, An \(O(\sqrt{n}L)\) iteration potential reduction algorithm for linear complementarity problems. Math. Program. 50, 331–342 (1991)

    Google Scholar 

  145. T.C. Koopmans, Optimum utilization of the transportation system, in Proceedings of the International Statistical Conference, Washington, DC, 1947

    Google Scholar 

  146. J. Kowalik, M.R. Osborne, Methods for Unconstrained Optimization Problems (Elsevier, New York, 1968)

    Google Scholar 

  147. H.W. Kuhn, The Hungarian method for the assignment problem. Naval Res. Logist. Q. 2, 83–97 (1955)

    Google Scholar 

  148. H.W. Kuhn, A.W. Tucker, Nonlinear programming, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, ed. by J. Neyman (University of California Press, Berkeley/Los Angeles, CA, 1961), pp. 481–492

    Google Scholar 

  149. C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1956)

    Google Scholar 

  150. J.B. Lasserre, Global optimization with polynomials and the problem of moments related. SIAM J. Optim. 11, 796–817 (2001)

    Google Scholar 

  151. M. Laurent, Matrix completion problems. Encycl. Optim. 3, 221–229 (2001)

    Google Scholar 

  152. E. Lawler, Combinatorial Optimization: Networks and Matroids (Holt, Rinehart, and Winston, New York, 1976)

    Google Scholar 

  153. C. Lemarechal, R. Mifflin, Nonsmooth optimization, in IIASA Proceedings III (Pergamon Press, Oxford, 1978)

    Google Scholar 

  154. C.E. Lemke, The dual method of solving the linear programming problem. Naval Res. Logist. Q. 1(1), 36–47 (1954)

    Google Scholar 

  155. E.S. Levitin, B.T. Polyak, Constrained minimization methods. Zh. vychisl. Math. Math. Fiz 6(5), 787–823 (1966)

    Google Scholar 

  156. M.S. Lobo, L. Vandenberghe, S. Boyd, Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998)

    Google Scholar 

  157. C. Loewner, Über monotone Matrixfunktionen. Math. Zeir. 38, 177–216 (1934). Also see C. Loewner, Advanced matrix theory, mimeo notes, Stanford University, 1957

    Google Scholar 

  158. F.A. Lootsma, Boundary properties of penalty functions for constrained minimization, Doctoral dissertation, Technical University, Eindhoven, May 1970

    Google Scholar 

  159. L. Lovász, A. Shrijver, Cones of matrices and setfunctions, and 0 − 1 optimization. SIAM J. Optim. 1, 166–190 (1990)

    Google Scholar 

  160. Z. Lu, L. Xiao, On the complexity analysis of randomized block-coordinate descent methods. Math. Program. (2013). doi: 10.1007/s10107-014-0800-2

    Google Scholar 

  161. D.G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969)

    Google Scholar 

  162. D.G. Luenberger, Hyperbolic pairs in the method of conjugate gradients. SIAM J. Appl. Math. 17(6), 1263–1267 (1969)

    Google Scholar 

  163. D.G. Luenberger, A combined penalty function and gradient projection method for nonlinear programming, Internal Memo, Department of Engineering-Economic Systems, Stanford University (June 1970)

    Google Scholar 

  164. D. G. Luenberger, The conjugate residual method for constrained minimization problems. SIAM J. Numer. Anal. 7(3), 390–398 (1970)

    Google Scholar 

  165. D.G. Luenberger, Control problems with kinks. IEEE Trans. Autom. Control AC-15(5), 570–575 (1970)

    Google Scholar 

  166. D.G. Luenberger, Convergence rate of a penalty-function scheme. J. Optim. Theory Appl. 7(1), 39–51 (1971)

    Google Scholar 

  167. D.G. Luenberger, The gradient projection method along geodesics. Manag. Sci. 18(11), 620–631 (1972)

    Google Scholar 

  168. D.G. Luenberger, Introduction to Linear and Nonlinear Programming, 1st edn. (Addison-Wesley, Reading, MA, 1973)

    Google Scholar 

  169. D.G. Luenberger, Linear and Nonlinear Programming, 2nd edn. (Addison-Wesley, Reading, MA, 1984)

    Google Scholar 

  170. D.G. Luenberger, An approach to nonlinear programming. J. Optim. Theory Appl. 11(3), 219–227 (1973)

    Google Scholar 

  171. Z.Q. Luo, W. Ma, A.M. So, Y. Ye, S. Zhang, Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27, 20–34 (2010)

    Google Scholar 

  172. Z.Q. Luo, J. Sturm, S. Zhang, Conic convex programming and self-dual embedding. Optim. Methods Softw. 14, 169–218 (2000)

    Google Scholar 

  173. I.J. Lustig, R.E. Marsten, D.F. Shanno, On implementing mehrotra’s predictor-corrector interior point method for linear programming. SIAM J. Optim. 2, 435–449 (1992)

    Google Scholar 

  174. N. Maratos, Exact penalty function algorithms for finite dimensional and control optimization problems, Ph.D. thesis, Imperial College Sci. Tech., University of London, 1978

    Google Scholar 

  175. G.P. McCormick, Optimality criteria in nonlinear programming, in Nonlinear Programming, SIAM-AMS Proceedings, vol. IX, 1976, pp. 27–38

    Google Scholar 

  176. L. McLinden, The analogue of Moreau’s proximation theorem, with applications to the nonlinear complementarity problem. Pac. J. Math. 88, 101–161 (1980)

    Google Scholar 

  177. N. Megiddo, Pathways to the optimal set in linear programming, in Progress in Mathematical Programming: Interior Point and Related Methods, ed. by N. Megiddo (Springer, New York, 1989), pp. 131–158

    Google Scholar 

  178. S. Mehrotra, On the implementation of a primal-dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992)

    Google Scholar 

  179. S. Mizuno, M.J. Todd, Y. Ye, On adaptive step primal-dual interior point algorithms for linear programming. Math. Oper. Res. 18, 964–981 (1993)

    Google Scholar 

  180. R.D.C. Monteiro, B.F. Svaiter, Iteration-complexity of block-decomposition algorithms and the alternating direction method of multipliers. SIAM J. Optim. 23, 475–507 (2013)

    Google Scholar 

  181. R.D.C. Monteiro, I. Adler, Interior path following primal-dual algorithms: part i: linear programming. Math. Program. 44, 27–41 (1989)

    Google Scholar 

  182. D.D. Morrison, Optimization by least squares. SIAM J. Numer. Anal. 5, 83–88 (1968)

    Google Scholar 

  183. B.A. Murtagh, Advanced Linear Programming (McGraw-Hill, New York, 1981)

    Google Scholar 

  184. B.A. Murtagh, R.W.H. Sargent, A constrained minimization method with quadratic convergence (Chap. 14), in Optimization, ed. by R. Fletcher (Academic, London, 1969)

    Google Scholar 

  185. K.G. Murty, Linear and Combinatorial Programming (Wiley, New York, 1976)

    Google Scholar 

  186. K.G. Murty, The Karmarkar’s algorithm for linear programming (Chap. 11.4.1), in Linear Complementarity, Linear and Nonlinear Programming. Sigma Series in Applied Mathematics, vol. 3 (Heldermann Verlag, Berlin, 1988), pp. 469–494

    Google Scholar 

  187. S.G., Nash, A. Sofer Linear and Nonlinear Programming (McGraw-Hill Companies, New York, 1996)

    Google Scholar 

  188. Y. Nesterov, Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM J. Optim. 22, 341–362 (2012)

    Google Scholar 

  189. Y.E. Nesterov, Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 9, 141–160 (1998)

    Google Scholar 

  190. Y. Nesterov, A method of solving a convex programming problem with convergence rate O((1∕k 2)). Soviet Math. Doklady 27(2), 372–376 (1983)

    Google Scholar 

  191. Y. Nesterov, M.J. Todd, Y. Ye, Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems. Math. Program. 84, 227–267 (1999)

    Google Scholar 

  192. Y. Nesterov, A. Nemirovskii, Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms (SIAM Publications, Philadelphia, 1994)

    Google Scholar 

  193. Y. Nesterov, M.J. Todd, Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22(1) 1–42 (1997)

    Google Scholar 

  194. Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course (Kluwer, Boston, 2004)

    Google Scholar 

  195. W. Orchard-Hays, Background development and extensions of the revised simplex method. RAND Report RM-1433, The RAND Corporation, Santa Monica, CA (1954)

    Google Scholar 

  196. A. Orden, Application of the simplex method to a variety of matrix problems, in Directorate of Management Analysis: “Symposium on Linear Inequalities and Programming”, ed. by A. Orden, L. Goldstein (DCS/Comptroller, Headquarters, U.S. Air Force, Washington, DC, 1952), pp. 28–50

    Google Scholar 

  197. A. Orden, The transshipment problem. Manag. Sci. 2(3), 276–285 (1956)

    Google Scholar 

  198. S.S. Oren, Self-scaling variable metric (ssvm) algorithms ii: implementation and experiments. Manag. Sci. 20, 863–874 (1974)

    Google Scholar 

  199. S.S. Oren, D.G. Luenberger, Self-scaling variable metric (ssvm) algorithms i: criteria and sufficient conditions for scaling a class of algorithms. Manag. Sci. 20, 845–862 (1974)

    Google Scholar 

  200. S.S. Oren, E. Spedicato, Optimal conditioning of self-scaling variable metric algorithms. Math. Program. 10, 70–90 (1976)

    Google Scholar 

  201. J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970)

    Google Scholar 

  202. C.C. Paige, M.A. Saunders, Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12(4), 617–629 (1975)

    Google Scholar 

  203. C. Papadimitriou, K. Steiglitz, Combinatorial Optimization Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, NJ, 1982)

    Google Scholar 

  204. P. Parrilo, Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96, 293–320 (2003)

    Google Scholar 

  205. G. Pataki, On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23, 339–358 (1998)

    Google Scholar 

  206. A. Perry, A modified conjugate gradient algorithm, Discussion Paper No. 229, Center for Mathematical Studies in Economics and Management Science, North-Western University, Evanston, IL (1976)

    Google Scholar 

  207. E. Polak, Computational Methods in Optimization: A Unified Approach (Academic, New York, 1971)

    Google Scholar 

  208. E. Polak, G. Ribiere, Note sur la Convergence de Methods de Directions Conjugres. Revue Francaise Informat. Recherche Operationnelle 16, 35–43 (1969)

    Google Scholar 

  209. M.J.D. Powell, An efficient method for finding the minimum of a function of several variables without calculating derivatives. Comput. J. 7, 155–162 (1964)

    Google Scholar 

  210. M.J.D. Powell, A method for nonlinear constraints in minimization problems, in Optimization, ed. by R. Fletcher Powell (Academic, London, 1969), pp. 283–298

    Google Scholar 

  211. M.J.D. Powell, On the convergence of the variable metric algorithm. Mathematics Branch, Atomic Energy Research Establishment, Harwell, Berkshire, England, (October 1969)

    Google Scholar 

  212. M.J.D. Powell, Algorithms for nonlinear constraints that use lagrangian functions. Math. Program. 14, 224–248 (1978)

    Google Scholar 

  213. B.N. Pshenichny, Y.M. Danilin, Numerical Methods in Extremal Problems (translated from Russian by V. Zhitomirsky) (MIR Publishers, Moscow, 1978)

    Google Scholar 

  214. M. Ramana, An exact duality theory for semidefinite programming and its complexity implications. Math. Program. 77, 129–162 (1997)

    Google Scholar 

  215. M. Ramana, L. Tuncel H. Wolkowicz, Strong duality for semidefinite programming. SIAM J. Optim. 7, 641–662 (1997)

    Google Scholar 

  216. J. Renegar, A polynomial-time algorithm, based on Newton’s method, for linear programming. Math. Program. 40, 59–93 (1988)

    Google Scholar 

  217. J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization (Society for Industrial and Applied Mathematics, Philadelphia, 2001)

    Google Scholar 

  218. R.T. Rockafellar, The multiplier method of hestenes and powell applied to convex programming. J. Optim. Theory Appl. 12, 555–562 (1973)

    Google Scholar 

  219. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970)

    Google Scholar 

  220. C. Roos, T. Terlaky, J.-Ph. Vial, Theory and Algorithms for Linear Optimization: An Interior Point Approach (Wiley, Chichester, 1997)

    Google Scholar 

  221. J. Rosen, The gradient projection method for nonlinear programming, I. Linear contraints. J. Soc. Ind. Appl. Math. 8, 181–217 (1960)

    Google Scholar 

  222. J. Rosen, The gradient projection method for nonlinear programming, II. Non-linear constraints. J. Soc. Ind. Appl. Math. 9, 514–532 (1961)

    Google Scholar 

  223. R. Saigal, Linear Programming: Modern Integrated Analysis (Kluwer Academic Publisher, Boston, 1995)

    Google Scholar 

  224. B. Shah, R. Buehler, O. Kempthorne, Some algorithms for minimizing a function of several variables. J. Soc. Ind. Appl. Math. 12, 74–92 (1964)

    Google Scholar 

  225. D.F. Shanno, Conditioning of quasi-Newton methods for function minimization. Math. Comput. 24, 647–656 (1970)

    Google Scholar 

  226. D.F. Shanno, Conjugate gradient methods with inexact line searches. Math. Oper. Res. 3(3) 244–2560 (1978)

    Google Scholar 

  227. A. Shefi, Reduction of linear inequality constraints and determination of all feasible extreme points, Ph.D. dissertation, Department of Engineering-Economic Systems, Stanford University, Stanford, CA, October 1969

    Google Scholar 

  228. W.F. Sheppard, On the calculation of the double integral expressing normal correlation. Trans. Camb. Philos. Soc. 19, 23–66 (1900)

    Google Scholar 

  229. M. Simonnard, Linear Programming, translated by William S. Jewell (Prentice-Hall, Englewood Cliffs, NJ, 1966)

    Google Scholar 

  230. M. Slater, Lagrange multipliers revisited: a contribution to non-linear programming. Cowles Commission Discussion Paper, Math 403 (November 1950)

    Google Scholar 

  231. A.M. So, Y. Ye, Theory of semidefinite programming for sensor network localization. Math. Program. 109, 367–384 (2007)

    Google Scholar 

  232. A.M. So, Y. Ye, J. Zhang, A unified theorem on SDP rank reduction. Math. Oper. Res. 33, 910–920 (2008)

    Google Scholar 

  233. G. Sonnevend, An ‘analytic center’ for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming, in System Modelling and Optimization: Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985, ed. by A. Prekopa, J. Szelezsan, B. Strazicky. Lecture Notes in Control and Information Sciences, vol. 84 (Springer, Berlin, 1986), pp. 866–876

    Google Scholar 

  234. G.W. Stewart, A modification of Davidon’s minimization method to accept difference approximations of derivatives. J. ACM 14, 72–83 (1967)

    Google Scholar 

  235. E.L. Stiefel, Kernel polynomials in linear algebra and their numerical applications. Nat. Bur. Stand. Appl. Math. Ser. 49, 1–22 (1958)

    Google Scholar 

  236. J.F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11&12, 625–633 (1999)

    Google Scholar 

  237. J. Sun, L. Qi, An interior point algorithm of \(O(\sqrt{n}\vert \ln (\epsilon )\vert )\) iterations for C 1-convex programming. Math. Program. 57, 239–257 (1992)

    Google Scholar 

  238. A. Tamir, Line search techniques based on interpolating polynomials using function values only. Manag. Sci. 22(5), 576–586 (1976)

    Google Scholar 

  239. K. Tanabe, Complementarity-enforced centered Newton method for mathematical programming, in New Methods for Linear Programming, ed. by K. Tone (The Institute of Statistical Mathematics, Tokyo, 1987), pp. 118–144

    Google Scholar 

  240. R.A. Tapia, Quasi-Newton methods for equality constrained optimization: equivalents of existing methods and new implementation, in Symposium on Nonlinear Programming III, ed. by O. Mangasarian, R. Meyer, S. Robinson (Academic, New York, 1978), pp. 125–164

    Google Scholar 

  241. M.J. Todd, A low complexity interior point algorithm for linear programming. SIAM J. Optim. 2, 198–209 (1992)

    Google Scholar 

  242. M.J. Todd, Y. Ye, A centered projective algorithm for linear programming. Math. Oper. Res. 15, 508–529 (1990)

    Google Scholar 

  243. K. Tone, Revisions of constraint approximations in the successive qp method for nonlinear programming problems. Math. Program. 26(2), 144–152 (1983)

    Google Scholar 

  244. D.M. Topkis, A note on cutting-plane methods without nested constraint sets. ORC 69-36, Operations Research Center, College of Engineering, Berkeley, CA (December 1969)

    Google Scholar 

  245. D.M. Topkis, A.F. Veinott Jr., On the convergence of some feasible direction algorithms for nonlinear programming. J. SIAM Control 5(2), 268–279 (1967)

    Google Scholar 

  246. J.F. Traub, Iterative Methods for the Solution of Equations (Prentice-Hall, Englewood Cliffs, NJ, 1964)

    Google Scholar 

  247. L. Tunçel, Constant potential primal-dual algorithms: a framework. Math. Prog. 66, 145–159 (1994)

    Google Scholar 

  248. R. Tutuncu, An infeasible-interior-point potential-reduction algorithm for linear programming, Ph.D. thesis, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1995

    Google Scholar 

  249. P. Tseng, Complexity analysis of a linear complementarity algorithm based on a Lyapunov function. Math. Program. 53, 297–306 (1992)

    Google Scholar 

  250. P.M. Vaidya, An algorithm for linear programming which requires \(O((m + n)n^{2} + (m + n)^{1.5}nL)\) arithmetic operations. Math. Prog. 47, 175–201 (1990). Condensed version in: Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, pp. 29–38

    Google Scholar 

  251. L. Vandenberghe, S. Boyd, Semidefinite programming. SIAM Rev. 38(1) 49–95 (1996)

    Google Scholar 

  252. R.J. Vanderbei, Linear Programming: Foundations and Extensions (Kluwer Academic Publishers, Boston, 1997)

    Google Scholar 

  253. S.A. Vavasis, Nonlinear Optimization: Complexity Issues (Oxford Science, New York, NY, 1991)

    Google Scholar 

  254. A.F. Veinott Jr., The supporting hyperplane method for unimodal programming. Oper. Res. XV 1, 147–152 (1967)

    Google Scholar 

  255. Y.V. Vorobyev, Methods of Moments in Applied Mathematics (Gordon and Breach, New York, 1965)

    Google Scholar 

  256. D.J. Wilde, C.S. Beightler, Foundations of Optimization (Prentice-Hall, Englewood Cliffs, NJ, 1967)

    Google Scholar 

  257. R.B. Wilson, A simplicial algorithm for concave programming, Ph.D. dissertation, Harvard University Graduate School of Business Administration, 1963

    Google Scholar 

  258. P. Wolfe, A duality theorem for nonlinear programming. Q. Appl. Math. 19, 239–244 (1961)

    Google Scholar 

  259. P. Wolfe, On the convergence of gradient methods under constraints. IBM Research Report RZ 204, Zurich (1966)

    Google Scholar 

  260. P. Wolfe, Methods of nonlinear programming (Chap. 6), in Nonlinear Programming, ed. by J. Abadie. Interscience (Wiley, New York, 1967), pp. 97–131

    Google Scholar 

  261. P. Wolfe, Convergence conditions for ascent methods. SIAM Rev. 11, 226–235 (1969)

    Google Scholar 

  262. P. Wolfe, Convergence theory in nonlinear programming (Chap. 1), in Integer and Nonlinear Programming, ed. by J. Abadie (North-Holland Publishing Company, Amsterdam, 1970)

    Google Scholar 

  263. S.J. Wright, Primal-Dual Interior-Point Methods (SIAM, Philadelphia, 1996)

    Google Scholar 

  264. G. Xue, Y. Ye, Efficient algorithms for minimizing a sum of Euclidean norms with applications. SIAM J. Optim. 7, 1017–1036 (1997)

    Google Scholar 

  265. Y. Ye, Approximating quadratic programming with bound and quadratic constraints. Math. Program. 84, 219–226 (1999)

    Google Scholar 

  266. Y. Ye, An O(n 3 L) potential reduction algorithm for linear programming. Math. Program. 50, 239–258 (1991)

    Google Scholar 

  267. Y. Ye, M.J. Todd, S. Mizuno, An \(O(\sqrt{n}L)\)-iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19, 53–67 (1994)

    Google Scholar 

  268. Y. Ye, Interior Point Algorithms (Wiley, New York, 1997)

    Google Scholar 

  269. Y. Ye, A new complexity result on solving the markov decision problem. Math. Oper. Res. 30, 733–749 (2005)

    Google Scholar 

  270. W.I. Zangwill, Nonlinear programming via penalty functions. Manag. Sci. 13(5), 344–358 (1967)

    Google Scholar 

  271. W.I. Zangwill, Nonlinear Programming: A Unified Approach (Prentice-Hall, Englewood Cliffs, NJ, 1969)

    Google Scholar 

  272. Y. Zhang, D. Zhang, On polynomiality of the mehrotra-type predictor-corrector interior-point algorithms. Math. Program. 68, 303–317 (1995)

    Google Scholar 

  273. G. Zoutendijk, Methods of Feasible Directions (Elsevier, Amsterdam, 1960)

    Google Scholar 

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Luenberger, D.G., Ye, Y. (2016). Basic Properties of Linear Programs. In: Linear and Nonlinear Programming. International Series in Operations Research & Management Science, vol 228. Springer, Cham. https://doi.org/10.1007/978-3-319-18842-3_2

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