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Topics on Solvability

  • George Isac
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 41)

Abstract

Given a complementarity problem, it is well known that its solution set can be empty. Because this fact, a natural question is: under what conditions a particular complementarity problem has at least a solution? This chapter is dedicated to the study of this problem. Here, we will discuss several aspects of solvability of complementarity problems and we and will present several general existence theorems, considered as classical results. Other existence theorems will be presented in Chapter 6–11.

Keywords

Variational Inequality Complementarity Problem Convex Cone Linear Complementarity Problem Nonlinear Complementarity Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. ABRAHAM, R. and ROBBIN, J. 1. TransversalMappings and Flows. Benjamin, New York (1967).MATHGoogle Scholar
  2. ALLEN, G. 1. Variational inequalities, complementarity problems and duality theorems. J. Math. Anal. 58 (1977), 1–10.MATHGoogle Scholar
  3. BAZARAA, M. S., GOODE, J. J. and NASHED, M. Z. 1. A nonlinear complementarity problem in mathematical programming in Banach spaces. Proc. Amer. Math. Soc. 35 Nr. 1 (1972), 165–170.MathSciNetMATHGoogle Scholar
  4. BERGE, G. 1. Topological Spaces. McMillan New York (1963).MATHGoogle Scholar
  5. BOD, P. 1. On closed sets having a least element. In: Optimization and Operations Research, (Eds. W. Oettli and K. Ritter), Lecture Notes in Economics and Math. Systems, Springer-Verlag, Nr. 117 (1976), 23–34.Google Scholar
  6. BOD, P. 2. Surun modèle non-linéare des rapports interindustriels, Rairo, Recherche Oper. 11 Nr. 4 (1977), 405–415.MathSciNetMATHGoogle Scholar
  7. BONNANS, J. F. and GONZAGA, C. C. 1. Convergence of interior point algorithms for the monotone linear complementarity problem. Math. Oper. Research, 21 Nr. 1 (1996), 1–25.MathSciNetMATHGoogle Scholar
  8. BORSUK, K. 1. Theory of Retracts. Warszawa (1967).MATHGoogle Scholar
  9. BORWEIN, J. M. 1. Alternative theorems and general complementarity problems. In: Infinite Programming, Lecture Notes in Econ. Math. Systems Nr. 259, Springer-Verlag (1985), 194–203.Google Scholar
  10. BORWEIN, J. M. and DEMPSTER, M. A. H. 1. The linear order complementarity problem. Math. Oper. Research 14, Nr. 3 (1989), 534–558.MathSciNetGoogle Scholar
  11. BOURBAKI, N. 1. Eléments de Mathématique, Topologie Générale. Chapitre 1 et Chapitre 2, Hermann, Paris (1965).Google Scholar
  12. BréZIS, H. 1. Equations et inéquations non-linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier 10 (1968), 115–176.Google Scholar
  13. BrÉZIS, H. and HESS, P. 1 Nonlinear mappings of nonotone type in Banach spaces. J. Funct. Anal., 11 (1972), 251–294.Google Scholar
  14. BROWDER, F. 1. On continuity of fixed points under deformation of contiuous mapings. Summa Brasil, Math. 4 (1960)Google Scholar
  15. CHANDRASEKARAN, R., KABADI, S. N. and SRIDHAR, R. 1. Integer solution for linear complentarity problem. Math. Oper. Research 23 Nr. 2, (1998), 390–402.MATHGoogle Scholar
  16. CHANDRASEKARAN, R. and SRIDHAR, R. 1. Integer solution for linear complentarity problem. Working paper Nr. 95 – 031, Faculty of Administration, Univ. of New Brunswick, Fredericton, Canada, (1995).Google Scholar
  17. COTTLE, R. W. 1. Note on a fundamental theorem in quadratic programming. Siam J. Appl. Math. 12 (1964), 663–665.MathSciNetMATHGoogle Scholar
  18. COTTLE, R. W. 2. Nonlinear programs with positively bounded Jacobians. Siam. J. Appl. Math. 14 Nr. 1 (1966), 147–158.MathSciNetMATHGoogle Scholar
  19. COTTLE, R. W. 3. Monotone solutions of the parametric linear complementarity problem. Math. Programming 3 (1972), 210–224.MathSciNetMATHGoogle Scholar
  20. COTTLE, R. W. and PANG, J. S. 1. A least-element theory of solving linear complementarity problems as linear programs. Math. Oper. Res. 3. Nr. 2 (1978), 155–170).MathSciNetMATHGoogle Scholar
  21. COTTLE, R. W. and PANG, J. S. 2. On solving linear complentarity problem as linear program. Math. Programming Study 7 (1978), 88–107.MathSciNetMATHGoogle Scholar
  22. COTTLE, R. W., PANG, J. S. and STONE, R.E. 1. The Linear Complementarity Problem. Academic Press (1992).MATHGoogle Scholar
  23. COTTLE, R. W. and STONE, R. E. 1. On the uniqueness of solutions to linear complementarity problems. Math. Programming 27 (1983), 191–213.MathSciNetMATHGoogle Scholar
  24. COTTLE, R. W. and VEINOTT, A. F. (Jr.) 1. Polyh edral s e ts having a least element. Math. Programming 3 (19 72), 238–249.Google Scholar
  25. COTTLE, R. W., VON RANDOW, R. and STONE, R. E. 1. On spherical convex sets and Q-matrices. Line ar Algebra Appl. 41 (1981), 73–80.MATHGoogle Scholar
  26. COTTLE, R. W. and YAO, J. C. 1. Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75 Nr. 2 (1992), 281–295.MathSciNetMATHGoogle Scholar
  27. DE LUCA, T., FACCHINEI, F. and KANZOW, C. 1. A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Programming 75 (1996), 407–439.MathSciNetMATHGoogle Scholar
  28. DORN, W. S. 1. Self-dual quadratic programs. Siam J. Appl. Math., 9 Nr. 1 (1961), 51–54.MathSciNetMATHGoogle Scholar
  29. EAVES, B. C 1. On the basic theorem of complementarity. Math. Programming 1 (1971), 68–75.MathSciNetMATHGoogle Scholar
  30. EILENBERG, S. and MONTGOMERY, D. 1. Fixed point theorems for multi-valued transformations. Amer. J. Math. 6; 8 (1946), 214–222.MathSciNetGoogle Scholar
  31. FACCHINEI, F. and KANZOW, C. 1. Beyond monotonicity in regularization methods for nonlinear complementarity problems. (Forthcoming, Siam J. Control Opt.).Google Scholar
  32. FAN, K. 1. A generalization of Tychonoff’s fixed point theorem. Math. Annalen 142 (1961), 305–310.MATHGoogle Scholar
  33. FAN, K. 2. A minimax inequality and applications. In: Inequalities III (Ed. O. Shisha), Academic Press (1972).Google Scholar
  34. FISCHER, A. 1. A specialNewton-type optimization methods. Optimization, 24 (1992), 269–284.MathSciNetMATHGoogle Scholar
  35. FISCHER, A. 2. On the local superlinear convergence of a Newton-type method for Lcp under weak conditions. Optimization Methods and Software 6 (1995), 83–107.Google Scholar
  36. FISCHER, A. 3. A Newton-type method for positive-semidefinite linear complementarity problems. J. Opt. Theory Appl. 86, Nr. 3 (1995), 585–608.MATHGoogle Scholar
  37. FISCHER, A. 4. Solution of monotone complementarity problems with locally Lipschitzian functions. Math. Programming 76 (1997), 513–532.MathSciNetMATHGoogle Scholar
  38. FISCHER, M. L. and TOLLE, J. W. 1. The nonlinear complementarity problem: existence and determination of solutions. Siam J. Control Opt. 15, Nr. 4 (1977), 612–624.Google Scholar
  39. FUKUSHIMA, M. 1. Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Programming 53 (1992), 99–110.MathSciNetMATHGoogle Scholar
  40. GALE, D. and NIKAIDO, H. 1. The Jacobian matrix and global univalence of mappings. Math. Ann. 159 (1965) 81–93.MathSciNetMATHGoogle Scholar
  41. GOULD, F. J. and TOLLE, J. W. 1. A unified approach to complementarity in optimization. J. Discrete Math. 7 (1974), 225–271.MathSciNetMATHGoogle Scholar
  42. GOWDA, M. S. 1. On Q-matrices. Math. Programming 49, (1990), 139–141.MathSciNetMATHGoogle Scholar
  43. GOWDA, M. S. 2. Complementarity problems over locally compact cones. Siam J. Control Opt., 27 Nr. 4 (1989), 836–841.MathSciNetMATHGoogle Scholar
  44. GOWDA M. S. and PANG, J. S. 1. The basic theorem of complementarity revisited. Math. Programming 58 (1993), 161–177.MathSciNetMATHGoogle Scholar
  45. GOWDA M. S. and PANG, J. S. 2. On the boundedness and stability of solutions to the affine variational inequality problem. Siam J. Control Opt., 32 Nr. 2 (1994), 421–441.MathSciNetMATHGoogle Scholar
  46. GOWDA M. S. and PANG, J. S. 3. Some existence results for multivalued complementarity problems. Math. Oper. Research 17, Nr. 3 (1992), 657–669.MathSciNetMATHGoogle Scholar
  47. GOWDA, M. S. and SEIDMAN, T. I. 1. Generalized linear complementarity problems. Math. Programming 46 (1990), 329–340.MathSciNetMATHGoogle Scholar
  48. GUO, J. S. and YAO, J. C. 1. The variational inequalities. Computers Math. Appl. 25, Nr. 3 (1993), 99–105.MathSciNetMATHGoogle Scholar
  49. HABETLER, G. I. and KOSTREVA, M. M. 1. Sets of generalized eomplementarity problems and P-matrices. Math. Oper. Research, 5 Nr. 2 (1980), 280–284.MathSciNetMATHGoogle Scholar
  50. HABETLER, G. J. and PRICE, A. L. 1. Existence theory for generalized nonlinear complementarity problems. J. Opt. Theory Appl., 7 Nr. 4 (1971), 223–239.MathSciNetMATHGoogle Scholar
  51. HERRERO, C. and SILVA, J. A. 1. On the equivalence between strong solvability and strict semimonotonicity for some systems involving Z-functions. Math. Programming 49, Nr. 3 (1991), 371–379.MathSciNetMATHGoogle Scholar
  52. HYERS, D. H., ISAC, G. and RASSIAS, TH. M. 1. Topics in Nonlinear Analysis and Applications. World Scientific, Singapore, New Jersey, London etc. (1997).MATHGoogle Scholar
  53. ISAC, G. 1. On some generalization of Karamardian’s theorem on the complementarity problem. Boll. U. M. I. (7) 2-B (1988), 323–332.Google Scholar
  54. ISAC, G. 2. Problèmes de Complémentarité (En Dimension Infinie). Publications du Département de Mathématiques et Informatique de l’Université de Limoges, France (1985).Google Scholar
  55. ISAC, G. 3. The numerical range theory and boundedness of solutions of the complementarity problem. J. Math. Anal. Appl. 143 Nr. 1 (1989), 235–251.MathSciNetGoogle Scholar
  56. ISAC, G. 4. A special variational inequality and the implicit complementarity problem. J. Fac. Sci. Univ. Tokyo Sect. Ia, Math. 37 (1990), 109–127.MathSciNetMATHGoogle Scholar
  57. ISAC, G. 5. Complementarity Problems. Lecture Notes in Mathematics, Springer-Verlag Nr. 1528, (1992).MATHGoogle Scholar
  58. ISAC, G. 6. Tihonov’s regularization and the complementarity problem in Hilbert spaces. J. Math. Anal. Appl. 174, Nr. 1 (1993). 53–66.MathSciNetMATHGoogle Scholar
  59. ISAC, G. and ThéRA, M. 1. A variational principle. Application to the nonlinear complementarity problem. In: Nonlinear and Convex Analysis (Proceedings in honor of Ky Fan), (Eds: B. L. Lin and S. Simons), Marcel Dekker (1987), 127–145.Google Scholar
  60. ISAC, G. and ThéRA, M. 2. Complementarity problem and the existence of the post-critical equilibrium state of a thin elastic plate. J. Opt. Theory Appl. 58, Nr. 2 (1988), 241–257.MATHGoogle Scholar
  61. ISAC, G., KOSTREVA, M. M. and WIECEK, M. M. 1. Multiple-objective approximation of feasible but unsolvable linear complementarity problems. J. Opt. Theory Appl. 86 Nr. 2 (1995), 389–405.MathSciNetMATHGoogle Scholar
  62. JAMESON, G. 1. Ordered Linear Spaces. Lecture Notes in Mathematics, Springer-Verlag Nr. 141, (1970).Google Scholar
  63. JOFFE, A. D. and TIHOMIROV, V. M. 1. Theory ofExtremal Problems. North-Holland, New York, (1979).Google Scholar
  64. KANEKO, I. 1. Linear complementarity problems and characterization of Minkowski matrices. Linear Algebra Appl. 20 (1978), 111–129.MATHGoogle Scholar
  65. KANEKO, I. 2. Isotone solutions of parametric linear complementarity problem. Math. Programming 12, (1977), 48–59.MathSciNetMATHGoogle Scholar
  66. KANEKO, I. 3. The number of solutions of a class of linear complementarity problems. Math. Programming, 17 (1979), 104–105.MathSciNetMATHGoogle Scholar
  67. KANEKO, I. 4. The number of solutions of a class of linear complementarity problems. W. P. 76–10, Department of Industrial Engineering, Univ. Wisconsin-Madison (Aug.-1976).Google Scholar
  68. KANZOW, C. 1. An inexact Qp-based mnethod for nonlinear complementarity problems. To appear in: Numerische Mathematik.Google Scholar
  69. KANZOW, C. and ZUPKE, M. 1. Inexact trust-region methods for nonlinear complementarity problems. To appear in: Reformulation, Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods. (Eds. M. Fukushima and L. Qi, Kluwer Academic Press).Google Scholar
  70. KARAMARDIAN, S. 1. The nonlinear complementarity problem with applications (I). J. Opt. Theory Appl. 4, Nr. 2 (1969), 87–98.MATHGoogle Scholar
  71. KARAMARDIAN, S. 2. The nonlinear complementarity problem with applications (Ii). J. Opt. Theory Appl. 4, Nr. 3 (1969), 167–181.MathSciNetMATHGoogle Scholar
  72. KARAMARDIAN, S. 3. Generalized complementarity problem. J. Opt. Theory Appl. 8, (1971), 161–168.MathSciNetMATHGoogle Scholar
  73. KARAMARDIAN, S. 4. The complementarity problem. Math. Programming 2, (1972), 107–129.MathSciNetMATHGoogle Scholar
  74. KARAMARDIAN, S. 5. Complementarity problems over cones with monotone and pseudomonotone maps. J. Opt. Theory Appl. 18, (1976), 445–454.MathSciNetMATHGoogle Scholar
  75. KELLY, L. M. and WATSON, L. T. 1. Q-matrices and spherieal geometry. Linear Algebra Appl. 25 (1979), 175–190.MathSciNetMATHGoogle Scholar
  76. KINDERLEHRER, D. and STAMPACCHIA, G. 1. An Introduction to Variational Inequalities and their Applications. Academic Press (1980).Google Scholar
  77. KOJIMA, A. 1. A unification of the existence theorems of the nonlinear complementarity problem. Math. Programming 9, (1975), 257–277.MathSciNetMATHGoogle Scholar
  78. KOJIMA, M. and SAIGAL, R. 1. On the number of solutions to a class of linear complementarity problems. Math. Programming 17 (1979), 136–139.MathSciNetMATHGoogle Scholar
  79. KOJIMA, M. and SAIGAL, R. 2. On the number of solutions to a class of complementarity problems. Math. Programming 21 (1981), 190–203.MathSciNetMATHGoogle Scholar
  80. KOJIMA, M., MIZUNO, S. and NOMA T. 1. A new continuation method for complementarity problems with uniform P-function. Math. Programming 43 (1989), 107–113.MathSciNetMATHGoogle Scholar
  81. KOJIMA, M., MEGIDDO, N. and NOMA, T. 1. Homotopie continuation method for nonlinear complementarity problems. Math. Oper. Research, 16 Nr. 4 (1991), 754–774.MathSciNetMATHGoogle Scholar
  82. KOSTREVA, M. M. 1. Finite test sets and P-matrices. Proc. Amer. Math. Soc. 84 Nr. 1 (1982), 104–105.MathSciNetMATHGoogle Scholar
  83. KOSTREVA, M. M. and WIECEK, M. M. 1. Linear complementarity problems and multiple objective programming. Math. Programming, 60 (1993), 349–359.MathSciNetMATHGoogle Scholar
  84. KOSTREVA, M. M. and YANG, X. Q. 1. Unified approaches for solvable and unsolvable linear complementarity problems. Preprint, (1999).Google Scholar
  85. KOSTREVA, M. M. and ZHENG, Q. 1. Integral global optimization method for solution of nonlinear complementarity problems. J. Global Optimization 5 (1994), 181–193.MathSciNetMATHGoogle Scholar
  86. KYPARISIS, J. 1. Uniqueness and differentiability of .solutions of parametric nonlinear complementarity problems. Math. Programming 36, (1986), 105–113.MathSciNetMATHGoogle Scholar
  87. LEMKE, C. E. 1. On complementarity pivot theory. In: Mathematics of Decision Sciences, (Eds. G. B. Dantzig and A. F. Veinott Jr.), Amer. Math. Soc. Providence R. I. (1968), 95–114.Google Scholar
  88. LLOYD, N. D. 1. Degree Theory, Cambridge University Press, Cambridge, London (1978).Google Scholar
  89. LUNA, G. 1. A remark on the complementarity problem. Proc. Amer. Math. Soc. 48, Nr. 1 (1975), 132–134.MathSciNetMATHGoogle Scholar
  90. LUO, Z, Q. MANGASARIAN, O. L., REN, J. and SOLODOV, M. V. 1. New error bounds for the linear complementarity problem. Math Oper. Research, 19 Nr. 4 (1994), 880–892.MathSciNetMATHGoogle Scholar
  91. MACLANE, S. 1. Homology. Academic Press (1963).MATHGoogle Scholar
  92. MANGASARIAN, O. L. 1. The ill-posed linear complementarity problem. In: Complementarity and Variational Problems. (Eds. Ferris, M. C. and Pang, J. S.), 226–233.Google Scholar
  93. MANGASARIAN, O. L. 2. Least norm solution of nonmonotone linear complementarity problems. In: Functional Analysis, Optimization and Mathematical Economics, (Ed. L. J. Leifman), Oxford University Press, (1990), 217–221.Google Scholar
  94. MANGASARIAN, O. L. 3. Some applications of penalty functions in mathematical programming. In: Optimization and Related Fields, (Eds. R. Conti, E. De Giorgi and F. Giannessi), Lecture Notes in Mathematics, Nr. 1190, Springer-Verlag, (1986), 307–329.Google Scholar
  95. MAS-COLELL, A. 1. A note on a theorem ofF. Browder. Math. Programming 6 (1974), 229–233.MathSciNetMATHGoogle Scholar
  96. MEGIDDO, N. 1. On the parametric nonlinear complementarity problem. Department of Statistics, Tel Aviv University. Tel Aviv (August 1975) (Revised).Google Scholar
  97. MEGIDDO, N. 2. On monotonicity in parametric linear complementarity problems. Math. Programming 12, Nr.1 (1977), 60–66.MathSciNetMATHGoogle Scholar
  98. MEGIDDO, N. 3. On the parametric nonlinear complementarity problem. Math. Programming Study 7, (1978), 142–150.MathSciNetMATHGoogle Scholar
  99. MEGIDDO, N. and KOJIMA, M. 1. On the existence and uniqueness of solutions in nonlinear complementarity theory. Math. Programming 12, (1977), 110–130.MathSciNetMATHGoogle Scholar
  100. MONTEIRO, R. D. C, PANG, J. S. and WANG, T. 1. A positive algorithm for the nonlinear complementarity problem. Siam J. Optimization, 5 Nr. 1 (1995), 129–148.MathSciNetMATHGoogle Scholar
  101. MORé, J. J. 1. Classes of functions and feasibility conditions in nonlinear complementarity problems. Math. Programming 6, (1974), 327–338.MathSciNetMATHGoogle Scholar
  102. MORé, J. J. 2. Coercivity conditions in nonlinear complementarity problems. Siam Review 16, Nr. 1, (1974), 1–16.MathSciNetMATHGoogle Scholar
  103. MORé, J. J. 3. Global methods for nonlinear complementarity problems. Math. Oper. Research 21, Nr. 3 (1996), 589–614.MATHGoogle Scholar
  104. MORé, J. J. and RHEINBOLDT, W. C. 1. On P-and S-function and related classes on n-dimensional nonlinear mappings. Linear Algebra Appl. 6 (1973), 45–68.MATHGoogle Scholar
  105. MOSCO, U. 1. Convergence of convex sets and solutions of variational inequalities. Adv. In Math. 3 (1969), 520–585.MathSciNetGoogle Scholar
  106. MURTY, K. G. 1. Linear Complementarity, Linear and Nonlinear Programming. Helderman Verlag, Berlin (1988).MATHGoogle Scholar
  107. NOOR, M. A. 1. An iterative scheme for a class of quasi-variational inequalities. J. Math. Anal. Appl. 110 (1985), 463–468.MathSciNetMATHGoogle Scholar
  108. NOOR, M. A. 2. Generalized complementarity problems. J. Math. Anal. Appl. 120 Nr. 1 (1986), 321–327.MathSciNetGoogle Scholar
  109. NOOR, M. A. 3. On the nonlinear complementarity problem. J. Math. Anal. Appl. 123 (1987), 455–460.MathSciNetMATHGoogle Scholar
  110. NOOR, M. A. 4. Fixed point approach for complementarity problems. J. Math. Anal. Appl. 133 (1988), 437–448.MathSciNetMATHGoogle Scholar
  111. NOOR, M. A. 5. Iterative methods for a class of complementarity problems. J. Math. Anal. Appl. 133 (1988), 366–382.MathSciNetMATHGoogle Scholar
  112. NOOR, M. A. 6. Wiener-Hopf equations and variational inequalities. J. Optim. Theory Appl. 79 (1993), 197–206.MathSciNetMATHGoogle Scholar
  113. NOOR, M. A. 7. Nonconvex functions and variational inequalities. J. Optim. Theory Appl. 86, (1995), 615–630.Google Scholar
  114. NOOR, M. A. and RASSIAS, TH. M. 1. Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993), 285–312.MathSciNetMATHGoogle Scholar
  115. NOOR, M. A., NOOR, K. I. and RASSIAS, TH. M. 1. Invitation to variational inequalities. In: Analysis, Geometry and Groups: A Reimann Legacy Volume, (Eds. Th. M. Rassias and H. M. Srivastava) Hadronic Press, Inc. Usa (1993), 373–448.Google Scholar
  116. ORTEGA, J. M. and RHEINBOLDT, W. 1. Iterative solutions of nonlinear equations in several variables. Academic Press, Inc. New York (1970).Google Scholar
  117. PALAIS, R. S. 1. Natural operations on differential forms. Trans. Amer. Math. Soc. 92, (1959), 125–141.MathSciNetMATHGoogle Scholar
  118. PANG, J. S. 1. On a class of least-element eomplementarity problems. Math. Programming 16 (1979), 111–126.Google Scholar
  119. PANG, J. S. 2. Complementarity problems. In: Handbook of Global Optimization, (Eds. R. Horst and P. M. Pardalos), Kluwer Academic Publishers (1995), 271–338.Google Scholar
  120. PARDALOS, P. M. and ROSEN, J. B. 1. Bounds for the solution set of linear complementarity problems. Discrete Appl. Math. 17 (1987), 255–261.MathSciNetMATHGoogle Scholar
  121. PARDALOS, P. M. and ROSEN, J. B. 2. Constraint Global Optimization: Algorithms and Applications. Lecture Notes in Computer Science, Springer-Verlag Nr. 268 (1987).Google Scholar
  122. PARDALOS, P. M. and NAGURNEY, A. 1. The integer linear complementarity problems. Intern. J. Computer Math. 31 (1990), 205–214.MATHGoogle Scholar
  123. PARK, S. 1. Generalized equilibrium problems and generalized complementarity problems. J. Opt. Theory Appl. 95 Nr. 2 (1997), 409–417.MATHGoogle Scholar
  124. ROBINSON, S. M. 1. Mathematical foundations of nonsmooth embedding methods. Math. Programming 48 (1990), 221–229.MathSciNetMATHGoogle Scholar
  125. ROBINSON, S. M. 2. Normal naps induced by linear transformations. Math. Oper. Research 17 Nr. 3 (1992), 691–714.MATHGoogle Scholar
  126. ROBINSON, S. M. 3. Homeomorphism conditions for normal maps of polyhedra. In: Optimization and Nonlinear Analysis (Eds. A. Joffe, M. Marcus and S. Reich), Longman, London (1992), 691–714.Google Scholar
  127. ROBINSON, S. M. 4. Nonsingularity and symmetry for linear normal maps. Math. Programming 62 (1993), 415–425.MathSciNetMATHGoogle Scholar
  128. ROBINSON, S. M. 5. Sensitivity analysis of variational inequalities by normal-map techniques. In: Variational Inequalities and Network Equilibrium Problems (Eds. F. Giannessi and A. Maugeri), Plenum Press, New York. (1995).Google Scholar
  129. ROCKAFELLAR, R. T. 1. Convex Analysis. Princeton University Press, Princeton, New Jersey (1970).MATHGoogle Scholar
  130. ROCKAFELLAR, R. T. 2. On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149 (1970), 75–88.MathSciNetMATHGoogle Scholar
  131. RHEINBOLDT, W. C. 1. On M-functions and their applications to nonlinear Gauss-Seidel iterations and to network flows. J. Math. Anal. Appl. 32 (1970), 274–307.MathSciNetMATHGoogle Scholar
  132. ROSEN, J. B. 1. Minimum norm solution to the linear complementarity problem. In: Functional Analysis, Optimization and Mathematical Economics (Ed. L. J. Leifman) Oxford University Press (1990), 208–216.Google Scholar
  133. SAIGAL, R. 1. Extension of the generalized complementarity problem. Math. Oper. Research, 1, Nr. 3 (1976), 260–266.MathSciNetMATHGoogle Scholar
  134. SAIGAL, R. and SIMON, C. B. 1. Generic properties of the complementarity problem. Math. Programming, 4 (1973), 324–335.MathSciNetMATHGoogle Scholar
  135. SMALE, S. 1. Global analysis in economics I. Pareto optimum and a generalization of Morse theory. In: Proceedings of the 1971 Dynamical Systems Symposium, Salvador, Brazil.Google Scholar
  136. STONE, R. E. 1. Linear complementarity problem with an invariant number of solutions. Math. Programming, 34 Nr. 3 (1986), 265–291.MathSciNetMATHGoogle Scholar
  137. SUBRAMANIAN, P. K. 1. A note on least two norm solutions of monotone complementarity problems. Appl. Math. Lett. 1 Nr. 4 (1988), 395–397.MathSciNetMATHGoogle Scholar
  138. TAMIR, A. 1. Minimality and complementarity properties associated to Z-functions and MfUnctions. Math. Programming 7, (1974), 17–31.MathSciNetMATHGoogle Scholar
  139. TAMIR, A. 2. On a characterization ofP-matrices. Math. Programming 4 (1973), 110–112.MathSciNetMATHGoogle Scholar
  140. TAMIR, A. 3. On the number of solutions to the linear complementarity problem. Math. Programming 10 (1976), 347–353.MathSciNetMATHGoogle Scholar
  141. VESCAN, R. T. 1. Un problème variationnel implicite faible. C. R. Acad. Sci. Paris, t. 299, Série A, Nr. 14 (1984), 655–658.Google Scholar
  142. VESCAN, R. T. 2. A weak implicit variational inequality. Preprint. Al I. Cuza University, Iasi (Romania), (1984)Google Scholar
  143. WATSON, L. T. 1. Some pertubation theorems for Q-matrices, Siam J. Appl. Math. 31 Nr.2 (1976), 379–384.MathSciNetMATHGoogle Scholar
  144. WEBER, V. H. 1. Φ-asymptotisches spectrum and surjektivitätssätze vom Fredholm type für nichtlineare operatoren mit anwendungen. Math. Machr. 117 (1984), 7–35.MATHGoogle Scholar
  145. WINTGEN, G. 1. Indifferente optimerungs probleme. Beitrag zur Internationalen Tagung. Mathematik und Kybernetik in der Okonomie. Berlin (1964), Konferenz-protokoll, Teill Ii (Akademie Verlag, Berlin, 3–6.Google Scholar
  146. WRIGHT, S. and RALPH, D. 1. A superlinear infeasible interior point algorithm for monotone complementarity problems. Math. Oper. Research, 21 Nr. 4 (1996), 815–838.MathSciNetMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • George Isac
    • 1
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada

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