Complementarity Problems

  • Jong-Shi Pang
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 2)

Abstract

This chapter presents a comprehensive treatment of the nonlinear complementarity problem and several related mathematical programs in finite dimensions. Topics discussed include existence theory, solution methods, sensitivity and stability analysis, and applications to equilibrium modeling and engineering problems. Some future research directions are suggested and an extensive list of references is given.

Key words:

Complementarity problems Variational inequalities Nonlinear programming Equilibrium modeling Global optimization Nonsmooth equations. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A1-Khayyal, F.A.: 1986, “Linear, quadratic, and bilinear programming approaches to the linear complementarity problem”, European Journal of Operational Research Vol. no. 86, pp. 216–227.Google Scholar
  2. 2.
    A1-Khayyal, F.A.: 1990, “On solving linear complementarity problems as bilinear programs”, Arabian Journal for Science and Engineering Vol. no. 15, pp. 639–646.Google Scholar
  3. 3.
    Anderson, E.J. and Wu, S.Y.: 1991, “The continuous complementarity problem”, Optimization Vol. no. 22, pp. 419–426.MathSciNetMATHGoogle Scholar
  4. 4.
    Aashtiani, H.Z. and Magnanti, T.L.: 1982, “Equilibria on a congested transportation network”, SIAM Journal on Algebraic and Discrete Methods Vol. no. 2, pp. 213–226.MathSciNetGoogle Scholar
  5. 5.
    Aubin, J.-P. and Frankowska, H.: 1990, Set-Valued Analysis, Birkäuser, Boston.MATHGoogle Scholar
  6. 6.
    Auclunuty, G.: 1989, “Variational principles for variational inequalities”, Numerical Functional Analysis and Optimization Vol. no. 10, pp. 863–874.Google Scholar
  7. 7.
    Balocchi, C. and Capelo, A.: 1984, Variational and Quasivariational Inequalities: Applications to free boundary problems, John Wiley and Sons, Chichester.Google Scholar
  8. 8.
    Bank, H., Guddat, J., Klatte, D., Kummer, B., and Tammer, K.: 1982, Non-Linear Parametric Optimization, Birkhäuser Berlag, Basel.Google Scholar
  9. 9.
    Björkman, G.: 1991, “The solution of large displacement frictionless contact problems using a sequence of linear complementarity problems”, International Journal for Numerical Methods in Engineering Vol. no. 31, pp. 1553–1556.MATHGoogle Scholar
  10. 10.
    Bonpans, J.F.: 1990, “Rates of convergence of Newton type methods for variational inequalities and nonlinear programming”, manuscript, INRIA.Google Scholar
  11. 11.
    Cao, M. and Ferris, M.C.: 1992, “An interior point algorithm for monotone affine variational inequalities”, Technical report, Computer Sciences Department, University of Wisconsin, Madison.Google Scholar
  12. 12.
    Cao, M. and Ferris, M.C.: 1992, “A pivotal method for affine variational inequalities”, Technical report #1114, Computer Sciences Department, University of Wisconsin, Madison.Google Scholar
  13. 13.
    Chan, D. and Pang, J.S.: 1982, “The generalized quasi-variational inequality problem”, Mathematics of Operations Research Vol. no. 7, pp. 211–222.MathSciNetMATHGoogle Scholar
  14. 14.
    Cheng, Y.C.: 1984, “On the gradient-projection method for solving the nonsymmetric linear complementarity problem”, Journal of Optimization Theory and Applications Vol. no. 43, pp. 527–541.MATHGoogle Scholar
  15. 15.
    Chipot, M.: 1984, Variational Inequalities and Flow in Porous Media, Applied Mathematical Sciences, Vol. no. 52, Springer Verlag, New York.Google Scholar
  16. 16.
    Corni, C. and Maier, G.: 1990, “Extremum theorem and convergence criterion for an iterative solution to the finite-step problem in elastoplasticity with mixed nonlinear hardening”, European Journal of Mechanics A/Solids Vol. no. 9, pp. 563–585.Google Scholar
  17. 17.
    Cottle, R.W.: 1964, “Nonlinear programs with positively bounded Jacobians”, Ph.D. thesis, Department of Mathematics, University of California, Berkeley.Google Scholar
  18. 18.
    Cottle, R.W.: 1966, “Nonlinear programs with positively bounded Jacobians”, SIA M Journal on Applied Mathematics Vol. no. 14, pp. 1472013158.MathSciNetMATHGoogle Scholar
  19. 19.
    Cottle, R.W., Kyparisis, J., and Pang, J.S. (eds.): 1990, Variational Inequality Problems, North-Holland, The Netherlands [also published as Mathematical Programming, Series B, Vol. no. 48 (2)].Google Scholar
  20. 20.
    Cottle, R.W., Kyparisis, J., and Pang, J.S. (eds.): 1990, Complementarity Problems, North-Holland, The Netherlands [also published as Mathematical Programming, Series B, Vol. no. 48 (3)].Google Scholar
  21. 21.
    Cottle, R.W., Pang, J.S., and Stone, R.E.: 1992, The Linear Complementarity Problem, Academic Press, Boston.MATHGoogle Scholar
  22. 22.
    Cottle, R.W., Pang, J.S., and Venkateswaran, V.: 1989, “Sufficient matrices and the linear complementarity problem”, Linear Algebra and its Applications Vol. no. 114/115, pp. 231–249.MathSciNetGoogle Scholar
  23. 23.
    Dafermos, S.C.: 1988: “Sensitivity analysis in variational inequalities”, Mathmematics of Operations Research Vol. no. 13, pp. 421–434.MathSciNetMATHGoogle Scholar
  24. 24.
    Dafermos, S.C. and McKelvey, S.C.: 1992, “Partitionable variational inequalities with applications to network and economic equilibria”, Journal of Optimization Theory and Applications Vol. no. 73, pp. 243–268.MathSciNetMATHGoogle Scholar
  25. 25.
    Dantzig, G.B. and Manne, A.S.: 1974, “A complementarity algorithm for an optimal capital path with invariant proportions”, Journal of Economic Theory Vol. no. 9, pp. 312–323.Google Scholar
  26. 26.
    De Donato, O. and Maier, G.: 1972, “Mathematical programming methods for the inelastic analysis of reinforced concrete frames allowing for limited rotation capacity”, International Journal for Numerical Methods in Engineering Vol. no. 4, pp. 307–329.Google Scholar
  27. 27.
    De Moor, B.: 1988, “Mathematical concepts and techniques for modelling of static and dynamic systems”, Ph.D. dissertation, Departement Elektrotechniek, Katholieke Universiteit Leuven, Leuven, Belgium.Google Scholar
  28. 28.
    Dennis, J.E., Jr. and Schnabel, R.B.: 1983, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, New Jersey.MATHGoogle Scholar
  29. 29.
    Dontchev, A.L. and Hager, W.W.: 1993, “Lipschitzian stability in nonlinear control and optimization”, SIAM Journal on Control and Optimization Vol. no. 31, pp. 569–603.MathSciNetMATHGoogle Scholar
  30. 30.
    Dontchev, A.L. and Hager, W.W.: 1992, “Implicit functions, Lipschitz maps, and stability in optimization”, manuscript, Department of Mathematics, University of Florida, Gainesville.Google Scholar
  31. 31.
    Eaves, B.C.: 1971, “On the basic theorem of complementarity”, Mathematical Programming Vol. no. 1, pp. 68–75.MathSciNetMATHGoogle Scholar
  32. 32.
    Eaves, B.C.: 1972, “Homotopies for computation of fixed points”, Mathematical Programming Vol. no. 3, pp. 1–22.MathSciNetMATHGoogle Scholar
  33. 33.
    Eaves, B.C.: 1976, “A short course in solving equations with PL homotopies”, in R.W. Cottle and C.E. Lemke, eds., Nonlinear Programming: SIAM-AMS Proceedings 9, American Mathematical Society, Providence, pp. 73–143.Google Scholar
  34. 34.
    Eaves, B.C.: 1983, “Where solving for stationary points by LCPs is mixing Newton iterates”, in B.C. Eaves, F.J. Gould, H.O. Peitgen, and M.J. Todd, eds., Homotopy Methods and Global Convergence, Plenum Press, New York, pp. 63–78.Google Scholar
  35. 35.
    Ferris, M.C. and Mangasarian, O.L.: 1991, “Error bounds and strong upper semicontinuity for monotone affine variational inequalities”, Technical report #1056, Computer Sciences Department, University of Wisconsin, Madison; Annals of Operations Research, forthcoming.Google Scholar
  36. 36.
    Fiacco, A.V.: 1983, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York.MATHGoogle Scholar
  37. 37.
    Fiacco, A.V. and McCormick, G.P.: 1968, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, New York.MATHGoogle Scholar
  38. 38.
    Florian, M.: 1989, Mathematical programming applications in national, regional and urban planning, in M. Iri and K. Tanabe, eds., Mathematical Programming: Recent Developments and Applications, Kluwer Academic Publishers, Tokyo, pp. 57–82.Google Scholar
  39. 39.
    Friedman, A.: 1982, Variational Principles and Free-Boundary Problems, John Wiley and Sons, New York.MATHGoogle Scholar
  40. 40.
    Friesz, T.L. and Harker, P.T.: 1985, “Freight network equilibrium: a review of the state of the art”, in A. Daughety, ed., Analytical Studies in Transportation Economics, Cambridge University Press, Cambridge, pp. 161–206.Google Scholar
  41. 41.
    Fujisawa, T. and Kuh, E.S.: 1972, “Piecewise-linear theory of resistive networks”, SIAM Journal on Applied Mathematics Vol. no. 22, pp. 307–328.MathSciNetMATHGoogle Scholar
  42. 42.
    Fukushima, M.: 1992, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems”, Mathematical Programming Vol. no. 53, pp. 45–62.MathSciNetGoogle Scholar
  43. 43.
    Gabriel, S.A. and Pang, J.S.: 1992, “An inexact NE/SQP method for solving the nonlinear complementarity problem”, Computational Optimization and Applications Vol. no. 1, pp. 67–91.MathSciNetMATHGoogle Scholar
  44. 44.
    Gabriel, S.A. and Pang, J.S.: 1993, “A trust region method for constrained nonsmooth equations”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
  45. 45.
    Gauvin, J.: 1977, “A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming”, Mathematical Programming Vol. no. 12, pp. 136–138.MathSciNetMATHGoogle Scholar
  46. 46.
    Garica, C.B. and Zangwill, W.I.: 1981, Pathways to Solutions, Fixed Points, and Equilibria, Prentice-Hall, Englewood Cliffs.Google Scholar
  47. 47.
    Gowda, M.S.: 1989, “ Pseucjomonotone and copositive-star matrices”, Linear Algebra and its Applications Vol. no. 113, pp. 107–110.MathSciNetMATHGoogle Scholar
  48. 48.
    Gowda, M.S.: 1989, “Complementarity problems over locally compact cones”, SIAM Journal on Control and Optimization Vol. no. 27, pp. 836–841.MathSciNetMATHGoogle Scholar
  49. 49.
    Gowda, M.S. and Pang, J.S.: 1992, “On solution stability of the linear complementarity problem”, Mathematics of Operations Research Vol. no. 17, pp. 77–83.MathSciNetMATHGoogle Scholar
  50. 50.
    Gowda, M.S. and Pang, J.S.: 1993, “Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory”, Mathematics of Operations Research, forthcoming.Google Scholar
  51. 51.
    Gowda, M.S. and Pang, J.S.: 1993, “On the boundedness and stability of solutions to the affine variational inequality problem”, SIAM Journal on Control and Optimization, forthcoming.Google Scholar
  52. 52.
    Gowda, M.S. and Seidman, T.I.: 1990, “Generalized linear complementarity problem”, Mathematical Programming Vol. no. 46, pp. 329–340.MathSciNetMATHGoogle Scholar
  53. 53.
    Gowda, M.S., and Sznajder, R.: 1993, “The generalized order linear complementarity problem”, SIAM Journal on Matrix Analysis and Applications Vol. no. 14, pp.Google Scholar
  54. 54.
    Guddat, J., Vasquez, F.G., with Jongen, H.Th.: 1990, Parametric Optimization: Singularities, Pathfollowing and Jumps, John Wiley & Sons, Chichester.Google Scholar
  55. 55.
    Güler, 0.: 1993, “Existence of interior points and interior paths in nonlinear complementarity problems”, Mathematics of Operations Research Vol. no. 18, pp. 128–147.MathSciNetMATHGoogle Scholar
  56. 56.
    Gupta, S. and Pardalos, P.M.: 1988, “A note on a quadratic formulation for linear complementarity problems”, Journal of Optimization Theory and Applications Vol. no. 57, pp. 197–201.MathSciNetMATHGoogle Scholar
  57. 57.
    Ha, C.D.: 1987, “Application of degree theory in stability of the complementarity problem”, Mathematics of Operations Research Vol. no. 31, 327–338.Google Scholar
  58. 58.
    Hansen, T. and Koopmans, T.C.: 1972, “On the definition and computation of a capital stock invariant under optimization”, Journal of Economic Theory Vol. no. 5, pp. 487–523.MathSciNetGoogle Scholar
  59. 59.
    Haraux, A.: 1977,“How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities”, Journal of the Mathematical Society of Japan Vol. no. 29, pp. 615–631.MathSciNetMATHGoogle Scholar
  60. 60.
    Harker, P.T.: 1987, Predicting Intercity Freight Flows, VNU Science Press, Utrecht.Google Scholar
  61. 61.
    Harker, P.T. and Pang, J.S.: 1990, “Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications”, Mathematical Programming, Series B Vol. no. 48, pp. 161–220.MathSciNetMATHGoogle Scholar
  62. 62.
    Harker, P.T. and Xiao, B: 1990, “Newton’s method for the nonlinear complementarity problem: a B-differentiable equation approach”, Mathematical Programming, Series B Vol. no. 48, pp. 339–357.MathSciNetMATHGoogle Scholar
  63. 63.
    Harrison, M.J. and Reiman, M.I.: 1981, “Reflected Brownian motion on an orthant”, The Journal of Probability Vol. no. 9, pp. 302–308.MathSciNetMATHGoogle Scholar
  64. 64.
    Hartman, P. and Stampacchia, G.: 1966, “On some nonlinear elliptic differential functional equations”, Acta Mathematica Vol. no. 15, pp. 153–188.Google Scholar
  65. 65.
    Hearn, D.W.: 1982, “The gap function of a convex program”, Operations Research Letters Vol. no. 1, pp. 67–71.MathSciNetMATHGoogle Scholar
  66. 66.
    Hlavâeek, I., Haslinger, J., News, J., and Lovisek, J.: 1988, Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences, Vol. no. 66, Springer-Verlag, New York.Google Scholar
  67. 67.
    Hoffman, A.J.: 1952, “On approximate solutions of systems of linear inequalities”, Journal of Research of the National Bureau of Standards Vol. no. 49, pp. 263–265.Google Scholar
  68. 68.
    Horst, R. and Tuy, H.: 1990, Global Optimization, Springer Verlag, New york.MATHGoogle Scholar
  69. 69.
    Ip, C.M. and Kyparisis, J., “Local convergence of quasi-Newton methods for B-differentiable equations”, Mathematical Programming Vol. no 56, pp. 71–90.Google Scholar
  70. 70.
    Isac, G.: 1992, Complementarily Problems, Lecture Notes in Mathematics, Springer Verlag, New York.Google Scholar
  71. 71.
    Isac, G. and Kostreva, M.M.: 1991, “The generalized order complementarity problem”, Journal of Optimization Theory and Applications Vol. no. 71, pp. 517–534.MathSciNetMATHGoogle Scholar
  72. 72.
    Istratescu, V.I.: 1981, Fixed Point Theory, D. Reidel Publishing Company, Boston.MATHGoogle Scholar
  73. 73.
    Jones, P.C.: 1977, “Calculation of an optimal invariant stock”, Ph.D. thesis, Department of Industrial Engineering, University of California, Berkeley.Google Scholar
  74. 74.
    Josephy, N.H.: 1979, “Newton’s method for generalized equations”, Technical report #1965, Mathematics Research Center, University of Wisconsin, Madison.Google Scholar
  75. 75.
    Josephy, N.H.: 1979, “Quasi-Newton methods for generalized equations”, Technical Report No. 1965, Mathematics Research Center, University of Wisconsin, Madison.Google Scholar
  76. 76.
    Kanzow, C.: 1993, “Some equation-based methods for the nonlinear complementarity problem”, Preprint 63, Institut far Angewandte Mathematik, Universität Hamburg, Hamburg, Germany.Google Scholar
  77. 77.
    Kanzow, C.: 1993, “Nonlinear complementarity as unconstrained optimization”, Preprint 67, Institut für Angewandte Mathematik, Universität Hamburg, Hamburg, Germany.Google Scholar
  78. 78.
    Karamardian, S.: 1969, “The nonlinear complementarity problem with applications, part 1”, Journal of Optimization Theory and Applications Vol. no. 4, pp. 87–98.MathSciNetMATHGoogle Scholar
  79. 79.
    Karamardian, S.: 1971, “Generalized complementarity problems”, Journal of Optimization Theory and Applications Vol. no. 19, pp. 161–168.MathSciNetGoogle Scholar
  80. 80.
    Karamardian, S.: 1972, “The complementarity problem”, Mathematical Programming Vol. no. 2, pp. 107–129.MathSciNetMATHGoogle Scholar
  81. 81.
    Kinderlehrer, D. and Stampacchia, G.: 1980, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York.MATHGoogle Scholar
  82. 82.
    King, A.J. and Rockafellâr, R.T.: 1992, “Sensitivity analysis for nonsmooth generalized equations”, Mathematical Programming Vol. no. 55, pp. 193–212.MathSciNetMATHGoogle Scholar
  83. 83.
    Klarbring, A. and Björkman, G.: 1992, “Solution of large displacement contact problems with friction using Newton’s method for generalized equations”, International Journal for Numerical Methods in Engineering Vol. no. 34, pp. 249–269.MATHGoogle Scholar
  84. 84.
    Kojima, M.: 1980, “Strongly stable stationary solutions in nonlinear programs”, in S.M. Robinson, ed., Analysis and Computation of Fixed Points, Academic Press, New York, pp. 93–138.Google Scholar
  85. 85.
    Kojima, M. and Hiabayashi, R.: 1984, “Continuous deformation of nonlinear programs”, Mathematical Programming Study Vol. no. 21, pp. 150–198.MATHGoogle Scholar
  86. 86.
    Kojima, M., Megiddo, N., and Noma, T.: 1991, “Homotopy continuation methods for nonlinear complementarity problems”, Mathematics of Operations Research Vol. no. 16, pp. 754–774.MathSciNetMATHGoogle Scholar
  87. 87.
    Kojima, M., Megiddo, N., Noma, T., and Yoshise, A.: 1991, A Unified Approach to Interior Point Algorithms for Linear Complementarily Problems, Lecture Notes in Computer Science 538, Springer Verlag, Berlin.Google Scholar
  88. 88.
    Kojima, M., Mizuno, S., and Noma, T.: 1989, “A new continuat:on method for complementarity problems with uniform P-functions”, Mathematical Programming Vol. no. 43, pp. 107–113.MathSciNetMATHGoogle Scholar
  89. 89.
    Kojima, M., Mizuno, S., and Noma, T.: 1990, “Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems”, Mathematics of Operations Research Vol. no. 15, pp. 662–675.MathSciNetMATHGoogle Scholar
  90. 90.
    Kojima, M., Noma, T., and Yoshise, A.: 1992, “Global convergence and detecting infeasibility in interior-point algorithms”, Research report B-257, Department of Information Sciences, Tokyo Institute of Technology, Tokyo.Google Scholar
  91. 91.
    Kostreva, M.M., 1978, “Block pivot methods for solving the complementarity problem”, Linear Algebra and its Applications Vol. no. 21, pp. 207–215.MathSciNetMATHGoogle Scholar
  92. 92.
    Kostreva, M.M.: 1984, “Elasto-hydrodynamic lubrication: a nonlinear complementarity problem”, International Journal for Numerical Methods in Fluids Vol. no. 4, pp. 377–397.MathSciNetMATHGoogle Scholar
  93. 93.
    Kyparisis, J.: 1985, “On uniqueness of Kuhn-Tucker multipliers in nonlinear programming”, Mathematical Programming Vol. no. 32, pp. 242–246.MathSciNetMATHGoogle Scholar
  94. 94.
    Kyparisis, J.: 1986, “Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems”, Mathematical Programming Vol. no. 36, pp. 105–113.MathSciNetMATHGoogle Scholar
  95. 95.
    Kyparisis, J.: 1990, “Sensitivity analysis for variational inequalities and nonlinear complementarity problems”, Annals of Operations Research Vol. no. 27, 143–174.MathSciNetMATHGoogle Scholar
  96. 96.
    Kyparisis, J.: 1990, “Solution differentiability for variational inequalities”, Mathematical Programming, Series B Vol. no. 48, pp. 285–302.MathSciNetMATHGoogle Scholar
  97. 97.
    Kyparisis, J.: 1992, “Parametric variational inequalities with multivalued solution sets”, Mathematics of Operations Vol. no. 17, pp. 341–364.MathSciNetMATHGoogle Scholar
  98. 98.
    Kyparisis, J. and Ip, C.M.: 1992, “Solution behavior for parametric implicit complementarity problems”, Mathematical Programming Vol. no. 56, pp. 71–90.MathSciNetMATHGoogle Scholar
  99. 99.
    Larsson, T. and Patriksson, M.: 1992, “A class of gap functions for variational inequalities”, manuscript, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden.Google Scholar
  100. 100.
    Lemke, C.E. and Howson, J.T., Jr.: 1964, “Equilibrium points of bimatrix games”, SIAM Journal on Applied Mathematics Vol. no. 12, pp. 413–423.MathSciNetMATHGoogle Scholar
  101. 101.
    Lin, Y.Y. and Pang, J.S.: 1987, “Iterative methods for large convex quadratic programs: A survey”, SIAM Journal on Control and Optimization Vol. no. 25, pp. 383–411.MathSciNetMATHGoogle Scholar
  102. 102.
    Lloyd, N.G.: 1978, Degree Theory, Cambridge University Press, Cambridge.Google Scholar
  103. 103.
    Lojasiewicz, M.S.: 1958, “Division d’une distribution par une fonction analytique de variables réelles”, Comptes de Rendus de Séance, Paris Vol. no. 146, pp. 683–686.MathSciNetGoogle Scholar
  104. 104.
    Lojasiewicz, M.S.: 1959, “Sur la problème de la division”, Studio Mathematica Vol. no. 18, pp. 87–136.MathSciNetMATHGoogle Scholar
  105. 105.
    Luo, X.D. and Luo, Z.Q.: 1992, “Extension of Hoffman’s error bound to polynomial systems”, Technical report, Communications Research Laboratory, McMaster University, Hamilton, Canada.Google Scholar
  106. 106.
    Luo, Z.Q.: 1992, “Convergence analysis of primal-dual interior point algorithms for solving convex quadratic programs”, Technical report, Communications Research Laboratory, McMaster University, Hamilton, Canada.Google Scholar
  107. 107.
    Luo, Z.Q., Mangasarian, O.L., Ren, J., and Solodov, M.V.: 1994, “New error bounds for the linear complementarity problem”, Mathematics of Operations Research Vol. no. 19, pp.Google Scholar
  108. 108.
    Luo, Z.Q. and Pang, J.S.: 1993, “Error bounds for analytic systems and their applications”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
  109. 109.
    Luo, Z.Q. and Tseng, P.: 1992: “Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem”,SIAM Journal on Optimization Vol. no. 2, pp. 43–54.MathSciNetMATHGoogle Scholar
  110. 110.
    Luo, Z.Q. and Tseng, P.: 1992: “On global error bound for a class of monotone affine variational inequality problems”, Operations Research Letters Vol. no. 11, pp. 159–165.MathSciNetMATHGoogle Scholar
  111. 111.
    Luo, Z.Q. and Tseng, P.: 1993, “Error bound and reduced gradient-projection algorithms for convex minimization over a polyhedral set”, SIAM Journal on Optimization Vol. no. 3, pp. 43–60.MathSciNetMATHGoogle Scholar
  112. 112.
    Lustig, I.J.: 1991, “Feasibility issues in a primal-dual interior-point method for linear programrning”, Mathematical Programming Vol. no. 49, pp. 145–162.MathSciNetGoogle Scholar
  113. 113.
    Magnanti, T.L.: 1984, “Models and algorithms for predicting urban traffic equilibrium”, in M. Florian, ed., Transportation Planning Models, North-Holland, Amsterdam, pp. 153–186.Google Scholar
  114. 114.
    Maier, G.: 1970, “A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes”, Meccanica Vol. no. 5, pp. 54–66.MATHGoogle Scholar
  115. 115.
    Maier, G. and Novati, G.: 1989, “A shakedown and bounding theory allowing for nonlinear hardening and second order geometric effects with reference to discrete structural models”, in J.A. König, ed., A Volumne in honor of A. Sawczuk.Google Scholar
  116. 116.
    Mandelbaum, A.: 1989, “The dynamic complementarity problem”, manuscript, Graduate School of Business, Stanford University, Stanford.Google Scholar
  117. 117.
    Mangasarian, O.L.: 1969, Nonlinear Programming, McGraw-Hill, New York.Google Scholar
  118. 118.
    Mangasarian, O.L.: 1976, “Equivalence of the complementarity problem to a system of nonlinear equations”, SIAM Journal on Applied Mathematics Vol. no. 31, pp. 89–992.MathSciNetMATHGoogle Scholar
  119. 119.
    Mangasarian, O.L.: 1980, “Locally unique solutions of quadratic programs, linear and nonlinear complementarity problems”, Mathematical Programming Vol. no. 19, pp. 200–212.MathSciNetMATHGoogle Scholar
  120. 120.
    Mangasarian, O.L.: 1992, “Global error bounds for monotone affine variational inequality problems”, Linear Algebra and its Applications Vol. no. 174, pp. 153–164.MathSciNetMATHGoogle Scholar
  121. 121.
    Mangasarian, O.L. and Fromovitz, S.: 1967, “The Fritz John optimality necessary conditions in the presence of equality and inequality constraints”, Journal of Mathematical Analysis and Applications Vol. no. 17, pp. 37–47.MathSciNetMATHGoogle Scholar
  122. 122.
    Mangasarian, O.L. and Shiau, T.H.: 1986, “Error bounds for monotone linear complementarity problems”, Mathematical Programming Vol. no. 36 pp. 81–89.MathSciNetMATHGoogle Scholar
  123. 123.
    Mangasarian, O.L. and Shiau, T.H.: 1986, “Error bounds for monotone linear complementarity problems”, Mathematical Programming Vol. no. 36 pp. 81–89.Google Scholar
  124. 124.
    Manne, A.S. (ed.): 1985, Economic Equilibrium: Model Formulation and Solution, North-Holland, Amsterdam [same as Mathematical Programming Study Vol. no. 231.Google Scholar
  125. 125.
    Marcotte, P.: 1985, “A new algorithm for solving variational inequalities, with application to the traffic assignment problem”, Mathematical Programming Vol. no. 33, pp. 339–351.MathSciNetGoogle Scholar
  126. 126.
    Marcotte, P. and Dussault, J.-P.: 1985, “A modified Newton method for solving variational inequalities”, Proceedings of the 24th IEEE Conference on Decision and Control, pp. 1433–1436.Google Scholar
  127. 127.
    Marcotte, P. and Dussault, J.-P.: 1987, “A note on a globally convergent Newton method for solving monotone variational inequalities”, Operations Research Letters Vol. no. 6, pp. 35–42.MathSciNetMATHGoogle Scholar
  128. 128.
    McCormick, G.P.: 1983, Nonlinear Programming: Theory, Algorithms, and Applications, John Wiley & Sons, New York.Google Scholar
  129. 129.
    McLinden, L.: 1980, “The complementarity problem for maximal monotone multifunction”, in R.W. Cottle, F. Giannessi, and J.L. Lions, eds., Variational Inequalities and Complementarity Problems, John Wiley, pp. 251–270.Google Scholar
  130. 130.
    Megiddo, N. and Kojima, M.: 1977, “On the existence and uniqueness of solutions in nonlinear complementarity problems”, Mathematical Programming Vol. no. 12, pp. 110–130.MathSciNetMATHGoogle Scholar
  131. 131.
    Minty, G.J.: 1962, “Monotone (non-linear) operators in Hilbert space”, Duke Mathematics Journal Vol. no. 29, pp. 341–346.MathSciNetMATHGoogle Scholar
  132. 132.
    Mittelman, H.D.: 1990, “Nonlinear parametrized equations: new results for variational problems and inequalities”, in E.L. Allgower and K. Georg, eds., Computational Solution of Nonlinear Systems of Equations, Lectures in Applied Mathematics, Vol. no. 26, American Mathematical Society, Providence, pp. 451–466.Google Scholar
  133. 133.
    Monteiro, R.D.C., Pang, J.S., and Wang, T.: 1992, “A positive algorithm for the nonlinear complementarity problem”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
  134. 135.
    Mordukhovich, B.: 1992, “Sensitivity analysis in nonsmooth optimization”, in D.A. Field & V. Komkov, eds., Theoretical Aspects of Industrial Design, SIAM Proceedings in Applied Mathematics, Vol. no. 58, pp. 32–46.Google Scholar
  135. 137.
    Mordukhovich, B.: 1992, “Lipschitzian stability of constraint systems and generalized equations”, manuscript, Department of Mathematics, Wayne State University, Detroit.Google Scholar
  136. 138.
    Moré, J. and Rheinboldt, W.C.: 1973, “On P- and S-functions and related class of n-dimensional nonlinear mappings”, Linear Algebra and its Applications Vol. no. 6, pp. 45–68.MATHGoogle Scholar
  137. 139.
    Murty, K.G.: 1988, Linear Complementarily, Linear and Nonlinear Programming, Helder-mann Verlag, Berlin.Google Scholar
  138. 140.
    Nagurney, A.: 1987, “Competitive equilibrium problems, variational inequalities and regional science”, Journal of Regional Science Vol. no. 27, pp. 55–76.Google Scholar
  139. 141.
    Nash, J.F.: 1950, “Equilibrium points in n-person games”, Proceedings of the National Academy of Sciences Vol. no. 36, pp. 48–49.MathSciNetMATHGoogle Scholar
  140. 142.
    Oh, K.P.: 1984, “The numerical solution of a dynamically loaded elastohydrodynamic contact as a nonlinear complementarity problem”, Transections of the ASME Vol. no. 106, pp. 88–94.Google Scholar
  141. 143.
    Oh, K.P.: 1986, “The formulation of the mixed lubrication problem as a generalized nonlinear complementarity problem”, Transactions of the ASME Vol. no. 108, pp. 598–604.Google Scholar
  142. 144.
    Ortega, J.M. and Rheinboldt, W.C.: 1970, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York.Google Scholar
  143. 145.
    Panagiotopoulos, P.D.: 1985, Inequality Problems in Mechanics and Applications, Birkhäuser, Boston.Google Scholar
  144. 146.
    Pang, J.S.: 1981, “The implicit complementarity problems”, in O.L. Mangasarian, R.R. Meyer, and S.M. Robinson, eds., Nonlinear Programming 4, Academic Press, New York, pp. 487–518.Google Scholar
  145. 147.
    Pang, J.S.: 1985, “Asymmetric variational inequality problems over product sets: applications and iterative methods”, Mathematical Programming Vol. no. 31, pp. 206–219.Google Scholar
  146. 148.
    Pang, J.S.: 1986, “Inexact Newton methods for the nonlinear complementarity problem”, Mathematical Programming Vol. no. 36, pp. 54–71.MATHGoogle Scholar
  147. 149.
    Pang, J.S.: 1987, “A posteriori error bounds for the linearly-constrained variational inequality problem”, Mathematics of Operations Research Vol. no. 12, pp. 474–484.Google Scholar
  148. 150.
    Pang, J.S.: 1990, “Newton’s method for B-differentiable equations”, Mathematics of Operations Research Vol. no. 15, pp. 311–341.MATHGoogle Scholar
  149. 151.
    Pang, J.S.: 1990, “Solution differentiability and continuation of Newton’s method for variational inequality problems over polyhedral sets”, Journal of Optimization Theory and Applications Vol. no. 66, pp. 121–135.MATHGoogle Scholar
  150. 152.
    Pang, J.S.: 1991, “A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity, and variational inequality problems”, Mathematical Programming Vol. no. 51, pp. 101–131.MATHGoogle Scholar
  151. 153.
    Pang, J.S.: 1992, “On local minima of nonlinear programs”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
  152. 154.
    Pang, J.S.: 1993, “Convergence of splitting and Newton methods for complementarity problems: an application of some sensitivity results”, Mathematical Programming Vol. no. 58, pp. 149–160.MATHGoogle Scholar
  153. 155.
    Pang, J.S.: 1993, “A degree-theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets”, Mathematical Programming, Series B, forthcoming.Google Scholar
  154. 156.
    Pang, J.S.: 1993, “Serial and parallel computations of Karush-Kuhn-Tucker points via non-smooth equations”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
  155. 157.
    Pang, J.S. and Chan, D.: 1982, “Iterative methods for variational and complementarity problems”, Mathematical Programming Vol. no. 24, pp. 284–313.MathSciNetMATHGoogle Scholar
  156. 158.
    Pang, J.S. and Gabriel, S.A.: 1993, “NE/SQP: A robust algorithm for the nonlinear complementarity problem”, Mathematical Programming, forthcoming.Google Scholar
  157. 159.
    Pang, J.S., Han, S.P., and Rangaraj, N.: 1991, “Minimization of locally Lipschitzian functions”, SIAM Journal of Optimization Vol. no. 1, pp. 57–82.MathSciNetMATHGoogle Scholar
  158. 160.
    Pang, J.S. and Qi, L.: 1993, “Nonsmooth equations: motivation and algorithms”, SIAM Journal on Optimization, Vol. no. 3, pp.Google Scholar
  159. 161.
    Pang, J.S. and Wang, Z.P.: 1990, “Embedding methods for variational inequality and nonlinear complementarity problems”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
  160. 162.
    Pang, J.S. and Yao, J.C.: 1992, “On a generalization of a normal map and equation”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
  161. 163.
    Pardalos, P.M. and Rosen, J.B.: 1987, Constrained Global Optimization: Algorithms and Applications, Springer Verlag, Berlin.Google Scholar
  162. 164.
    Pardalos, P.M. and Rosen, J.B.: 1988, “Global optimization approach to the linear complementarity problem”, SIAM Journal on Scientific and Statistical Computations Vol. no. 9, pp. 341–353.MathSciNetMATHGoogle Scholar
  163. 165.
    Qi, L.: 1993, “Convergence analysis of some algorithms for solving nonsmooth equations”, Mathematics of Operations Research Vol. no. 18, pp. 227–244.MATHGoogle Scholar
  164. 166.
    Qi, L. and Jiang, H.: 1992, “On the range sets of variational inequalities”, manuscript, Department of Applied Mathematics, University of New South Wales, Kensington, New South Wales, Australia.Google Scholar
  165. 167.
    Qi, L. and Sun, J.: 1993, “A nonsmooth version of Newton’s method”, Mathematical Programming Vol. no. 58, pp. 353–368.MathSciNetMATHGoogle Scholar
  166. 168.
    Qiu, Y. and Magnanti, T.: 1989, “Sensitivity analysis for variational inequalities defined on polyhedral sets”, Mathematics of Operations Research Vol. no. 14, pp. 410–332.MathSciNetMATHGoogle Scholar
  167. 169.
    Qiu, Y. and Magnanti, T.: 1992, “Sensitivity analysis for variational inequalities”, Mathematics of Operations Research Vol. no. 17, pp. 61–76.MathSciNetMATHGoogle Scholar
  168. 170.
    Ralph, D.: 1993, “Global convergence of damped Newton’s method for nonsmooth equations, via the path search”, Mathematics of Operations Research, forthcoming.Google Scholar
  169. 171.
    Reiman, M.I. and Williams, R.J.: 1988, “A boundary property of semi-martingale reflecting Brownian motions”, Probability Theory and Related Fields Vol. no. 77, pp. 87–97.MathSciNetMATHGoogle Scholar
  170. 172.
    Reinoza, A.: 1985, “The strong positivity conditions”, Mathematics of Operations Research Vol. no. 10, pp. 54–62.MathSciNetMATHGoogle Scholar
  171. 173.
    Robinson, S.M.: 1979, “Generalized equations and their applications, part I: basic theory”, Mathematical Programming Study Vol. no. 10, pp. 128–141.MATHGoogle Scholar
  172. 174.
    Robinson, S.M.: 1980, “Strongly regular generalized equations”, Mathematics of Operations Research Vol. no. 5, pp. 43–62.MATHGoogle Scholar
  173. 175.
    Robinson, S.M.: 1981, “Some continuity properties of polyhedral multifunction”, Mathematical Programming- Study Vol. no. 14, pp. 206–214.Google Scholar
  174. 176.
    Robinson, S.M.: 1982, “Generalized equations and their applications, part II: applications to nonlinear programming”, Mathematical Programming Study Vol. no. 19, pp. 200–221.MATHGoogle Scholar
  175. 177.
    Robinson, S.M.: 1983, “Generalized equations”, in A. Bachem, M. Grötschel, and B. Korte, eds., Mathematical Programming: The State of the Art, Springer Verlag, Berlin, pp. 346–367.Google Scholar
  176. 178.
    Robinson, S.M.: 1985, “Implicit B-differentiability in generalized equations”, Technical report #2854, Mathematics Research Center, University of Wisconsin, Madison.Google Scholar
  177. 179.
    Robinson, S.M.: 1987, “Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity”, Mathematical Programming Study, Vol. no. 30, pp. 45–66. Corrigenda, Mathematical Programming Vol. no. 49, pp. 143.Google Scholar
  178. 180.
    Robinson, S.M.: 1988, “Newton’s method for a class of nonsmooth functions”, manuscript, Department of Industrial Enginering, University of Wisconsin, Madison.Google Scholar
  179. 181.
    Robinson, S.M.: 1990, “Mathematical foundations of embedding methods for nonsmooth equations”, Mathematical Programming, Series B Vol. no. 48, pp. 221–230.MATHGoogle Scholar
  180. 190.
    Robinson, S.M.: 1990, “An implicit-function theorem for a class of nonsmooth functions”, Mathematics of Operations Research Vol. no. 16, pp. 292–309.Google Scholar
  181. 182.
    Robinson, S.M.: 1992, “Normal maps induced by linear transformations”, Mathematics of Operations Research Vol. no. 17, pp. 691–714.MATHGoogle Scholar
  182. 183.
    Robinson, S.M.: 1992, “Homeomorphism conditions for normal maps of polyhedra”, in A. Ioffe, M. Marcus, and S. Reich, eds., Optimization and Nonlinear Analysis, Longman, London, pp. 691–714.Google Scholar
  183. 184.
    Rockafellar, R.T.: 1970, Convex Analysis, Princeton University Press, Princeton.Google Scholar
  184. 185.
    Rockafellar, R.T.: 1987, “Linear-quadratic programming and optimal control”, SIAM Journal on Control and Optimization Vol. no. 25, pp. 781–814.MathSciNetMATHGoogle Scholar
  185. 186.
    Rockafellar, R.T.: 1989, “Proto-differentiability of set-valued mappings and its applications in optimization”, in H. Attouch, J.-P. Aubin, F.H. Clarke, and I. Ekeland, eds., Analyse Non Linéaire, Gauthier-Villars, Paris, pp. 449–482.Google Scholar
  186. 187.
    Rockafellar, R.T.: 1990, “Computational schemes for solving large-scale problems in extended linear-quadratic programming”, Mathematical Programming Vol. no. 48, pp. 447–474.MathSciNetMATHGoogle Scholar
  187. 188.
    Rockafellar, R.T.: 1990, “Computational schemes for solving large-scale problems in extended linear-quadratic programming”, Mathematical Programming Vol. no. 48, pp. 447–474.Google Scholar
  188. 189.
    Rockafellar, R.T. and Wets, R.J-B.: 1986, “A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming”, Mathematical Programming Studies Vol. no. 28, pp. 63–93.MathSciNetMATHGoogle Scholar
  189. 190.
    Rockafellar, R.T. and Wets, R.J-B.: 1987, “Linear-quadratic problems with stochastic penalties: the finite generation algorithm”, in Y. Ermoliev and R.J-B.Wets, eds., Numerical Techniques for Stochastic Optimization Problems, Springer-Verlag Lecture Notes in Control and Information Sciences Vol. no. 81, pp. 545–560.Google Scholar
  190. 200.
    Rockafellar, R.T. and Wets, R.J-B.: 1990, “Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time”, SIAM Journal on Control and Optimization Vol. no. 28, pp. 810–822.MathSciNetMATHGoogle Scholar
  191. 191.
    Scarf, H. (with the collaboration of T. Hansen): 1973, The Computation of Economic Equilibria, Yale University Press, New Haven.MATHGoogle Scholar
  192. 192.
    Stevens, S.N. and Lin, S.M.: 1981, “Analysis of piecewise linear resistive networks using complementarity pivot theory”, IEEE Transactions on Circuits and Systems Vol. no. CAS-28, pp. 429–441.MathSciNetGoogle Scholar
  193. 193.
    Stevens, S.N. and Lin, S.M.: 1981, “Analysis of piecewise linear resistive networks using complementarity pivot theory”, IEEE Transactions on Circuits and Systems Vol. no. CAS-28, pp. 429–441.Google Scholar
  194. 194.
    Sun., M.: 1987, “Singular control problems in bounded intervals”, Stochastics Vol. no. 21, pp. 303–344.MATHGoogle Scholar
  195. 195.
    Sun., M.: 1989, “Monotonicity of Mangasarian’s iterative algorithm for generalized linear complementarity problems”, Journal of Mathematical Analysis and Applications Vol. no. 144, pp. 474–485.MATHGoogle Scholar
  196. 196.
    Taji, K., Fukushima, M., and Ibaraki, T.: 1993, “A globally convergent Newton method for solving monotone variational inequalities”, Mathematical Programming Vol. no. 58, pp. 369–384.MathSciNetMATHGoogle Scholar
  197. 197.
    Thompson, G.L. and Thore, S.: 1991, “Economic disequilibrium by mathematical programming”, Journal of Optimization Theory and Applications Vol. no. 71, pp. 169–187.MathSciNetMATHGoogle Scholar
  198. 198.
    Thore, S., Nagurney, A., and Pan, J.: 1992, “Generalized goal programming and variational inequalities”, Operations Research Letters Vol. no. 12, pp. 217–226.MathSciNetMATHGoogle Scholar
  199. 199.
    Tin-Loi, F. and Pang, J.S.: 1993, “Elastoplastic analysis of structures with nonlinear hardening: a nonlinear complementarity approach”, Computer Methods in Applied Mechanics and Engineering, forthcoming.Google Scholar
  200. 200.
    Tobin, R.L.: 1986: “Sensitivity analysis for variational inequalities”, Journal of Optimization Theory and Applications Vol. no. 48, pp. 191–204.MathSciNetMATHGoogle Scholar
  201. 201.
    Todd, M.J.: 1976, Computation of Fixed Points and Applications, Lecture Notes in Economics and Mathematical Systems 124, Springer-Verlag, Heidelberg.Google Scholar
  202. 202.
    Todd, M.J. and Ye, Y.: 1990, “A centered projective algorithm for linear programming”, Mathematics of Operations Research Vol. no. 15, pp. 508–529.MathSciNetMATHGoogle Scholar
  203. 203.
    Vandenberghe, L., De Moor, B., and Vandewalle, J.: 1989, “The generalized linear complementarity problem problem applied to the complete analysis of resistive piecewise-linear circuits”, IEEE Transactions on Circuits and Systems Vol. no. 36, pp. 1382–1391.Google Scholar
  204. 204.
    Van Eijndhoven, J.T.L.: 1986, “Solving the linear complementarity problem in circuit simulation”, SIAM Journal on Control and optimization Vol. no. 24, pp. 1050–1062.MATHGoogle Scholar
  205. 205.
    Wright, S.: 1992, “An infeasible interior point algorithm for linear complementarity problems”, Technical report MCS-P331–1092, Mathematics and Computer Science Division, Argonne National Laboratory.Google Scholar
  206. 206.
    Wu, J.H.: 1992, “On descent simplicial decomposition methods for the monotone variational inequality problem with its application to the network equilibrium problem”, manuscript, Centre de Recherche sur les Transports, Université de Montréal, Montréal.Google Scholar
  207. 207.
    Wu, J.H., Florian, M., and Marcotte, P.: 1991, “A general descent framework for the mono-tone variational inequality problem”, Publication 723, Centre de Recherche sur les Transports, Université de Montréal, Montréal.Google Scholar
  208. 208.
    Yao, J.C.: 1990, “Generalized quasi variational inequality and implicit complementarity problems”, Ph.D. dissertation, Department of Operations Research, Stanford University, Stanford.Google Scholar
  209. 209.
    Yao, J.C.: 1991, “The generalized quasi-variational inequality problem with applications”, Journal of Mathematical Analysis and Applications Vol. no. 158, pp. 139–160.MATHGoogle Scholar
  210. 210.
    Yau, S.T. and Gao, Y.: 1992, “Obstacle problem for von K .rmân equations”, Advances in Applied Mathematics Vol. no. 13, pp. 123–141.MathSciNetMATHGoogle Scholar
  211. 211.
    Zarantonello, E.H.: 1971, “Projections on convex sets in Hilbert space and spectral theory”, in E.H. Zarantonello, ed., Contributions to Nonlinear Functional Analysis, Academic Press, New York, pp. 237–424.Google Scholar
  212. 212.
    Zhang, Y.: 1994, “On the convergence of an infeasible interior-point algorithm for linear programming and other problems”, SIAM Journal on Optimization Vol. no. 4, pp.Google Scholar
  213. 213.
    Zhu, C.Y.: 1992, “Modified proximal point algorithm for extended linear-quadratic programming”, Computational Optimization and Applications Vol. no. 2, pp. 182–205.Google Scholar
  214. 214.
    Zhu, C.Y. and Rockafellar, R.T.: 1993, “Primal-dual projected gradient algorithms for extended linear-quadratic progranuning”, SIAM Journal on Optimization Vol. no. 3, pp.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Jong-Shi Pang
    • 1
  1. 1.Department of Mathematical SciencesThe Johns Hopkins UniversityBaltimoreUSA

Personalised recommendations