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Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications

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Abstract

Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.

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References

  1. H.Z. Aashtiani and T.L. Magnanti, “Equilibria on a congested transportation network,”SIAM Journal on Algebraic and Discrete Methods 2 (1981) 213–226.

    Google Scholar 

  2. H.Z. Aashtiani and T.L. Magnanti, “A linearization and decomposition algorithm for computing urban traffic equilibria,”Proceedings of the 1982IEEE International Large Scale Systems Symposium (1982) 8–19.

  3. M. Abdulaal and L.J. LeBlanc, “Continuous equilibrium network design models,”Transportation Research 13B (1979) 19–32.

    Google Scholar 

  4. M. Aganagic, “Variational inequalities and generalized complementarity problems,” Technical Report SOL 78-11, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA 1978).

    Google Scholar 

  5. B.H. Ahn,Computation of Market Equilibria for Policy Analysis: The Project Independence Evaluation Study (PIES) Approach (Garland, NY, 1979).

  6. B.H. Ahn, “A Gauss-Seidel iteration method for nonlinear variational inequality problems over rectangles,”Operations Research Letters 1 (1982) 117–120.

    Google Scholar 

  7. B.H. Ahn, “A parametric network method for computing nonlinear spatial equilibria,” Research report, Department of Management Science, Korea Advanced Institute of Science and Technology (Seoul, Korea, 1984).

    Google Scholar 

  8. B.H. Ahn and W.W. Hogan, “On convergence of the PIES algorithm for computing equilibria,”Operations Research 30 (1982) 281–300.

    Google Scholar 

  9. E. Allgower and K. Georg, “Simplicial and continuation methods for approximating fixed points and solutions to systems of equations,”SIAM Review 22 (1980) 28–85.

    Google Scholar 

  10. R. Asmuth, “Traffic network equilibrium,” Technical Report SOL 78-2, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1978).

    Google Scholar 

  11. R. Asmuth, B.C. Eaves and E.L. Peterson, “Computing economic equilibria on affine networks with Lemke's algorithm,”Mathematics of Operations Research 4 (1979) 207–214.

    Google Scholar 

  12. J.P. Aubin,Mathematical Methods of Game and Economic Theory (North-Holland, Amsterdam, 1979).

    Google Scholar 

  13. M. Avriel,Nonlinear Programming: Analysis and Methods (Prentice-Hall, Englewood Cliffs, NJ, 1976).

    Google Scholar 

  14. S.A. Awoniyi and M.J. Todd, “An efficient simplicial algorithm for computing a zero of a convex union of smooth functions,”Mathematical Programming 25 (1983) 83–108.

    Google Scholar 

  15. C. Baiocchi and A. Capelo,Variational and Quasivariational Inequalities: Application to Free-Boundary Problems (Wiley, New York, 1984).

    Google Scholar 

  16. B. Banks, J. Guddat, D. Klatte, B. Kummer and K. Tammer,Nonlinear Parametric Optimization (Birkhauser, Basel, 1983).

    Google Scholar 

  17. V. Barbu,Optimal Control of Variational Inequalities (Pitman Advanced Publishing Program, Boston, 1984).

    Google Scholar 

  18. M.J. Beckman, C.B. McGuire, and C.B. Winston,Studies in the Economics of Transportation (Yale University Press, New Haven, CT, 1956).

    Google Scholar 

  19. A. Bensoussan, “Points de Nash dans le cas de fonctionnelles quadratiques et jeux differentials linéaires aN personnes,”SIAM Journal on Control 12 (1974) 460–499.

    Google Scholar 

  20. A. Bensoussan, M. Goursat and J.L. Lions, “Contrôle impulsionnel et inéquations quasivariationnelles stationnaires,”Comptes Rendus Academie Sciences Paris 276 (1973) 1279–1284.

    Google Scholar 

  21. A. Bensoussan and J.L. Lions, “Nouvelle formulation de problèmes de contrôle impulsionnel et applications,”Comptes Rendus Academie Sciences Paris 276 (1973) 1189–1192.

    Google Scholar 

  22. A. Bensoussan and J.L. Lions, “Nouvelles méthodes en contrôle impulsionnel,”Applied Mathematics and Optimization 1 (1974) 289–312.

    Google Scholar 

  23. C. Berge,Topological Spaces (Oliver and Boyd, Edinburgh, Scotland, 1963).

    Google Scholar 

  24. D.P. Bertsekas and E.M. Gafni, “Projection methods for variational inequalities with application to the traffic assignment problem,”Mathematical Programming Study 17 (1982) 139–159.

    Google Scholar 

  25. K.C. Border,Fixed Point Theorems with Applications to Economics and Game Theory (Cambridge University Press, Cambridge, 1985).

    Google Scholar 

  26. F.E. Browder, “Existence and approximation of solutions of nonlinear variational inequalities,”Proceeding of the National Academy of Sciences, U.S.A. 56 (1966) 1080–1086.

    Google Scholar 

  27. M. Carey, “Integrability and mathematical programming models: a survey and parametric approach,”Econometrica 45 (1977) 1957–1976.

    Google Scholar 

  28. D. Chan and J.S. Pang, “The generalized quasi-variational inequality problem,”Mathematics of Operations Research 7 (1982) 211–222.

    Google Scholar 

  29. G.S. Chao and T.L. Friesz, “Spatial price equilibrium sensitivity analysis,”Transportation Research 18B (1984) 423–440.

    Google Scholar 

  30. S.C. Choi, W.S. DeSarbo and P.T. Harker, “Product positioning under price competition,”Management Science 36 (1990) 265–284.

    Google Scholar 

  31. R.W. Cottle,Nonlinear Programs with Positively Bounded Jacobians. Ph.D. dissertation, Department of Mathematics, University of California (Berkeley, CA, 1964).

    Google Scholar 

  32. R.W. Cottle, “Nonlinear programs with positively bounded Jacobians,”SIAM Journal on Applied Mathematics 14 (1966) 147–158.

    Google Scholar 

  33. R.W. Cottle, “Complementarity and variational problems,”Symposia Mathematica XIX (1976) 177–208.

    Google Scholar 

  34. R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming,”Linear Algebra and Its Applications 1 (1968) 103–125.

    Google Scholar 

  35. R.W. Cottle, F. Giannessi and J.L. Lions, eds.,Variational Inequalities and Complementarity Problems: Theory and Applications (Wiley, New York, 1980).

    Google Scholar 

  36. R.W. Cottle, G.J. Habetler and C.E. Lemke, “Quadratic forms semi-definite over convex cones,” in: H.W. Kuhn, ed.,Proceedings of the Princeton Symposium on Mathematical Programming (Princeton University Press, Princeton, NJ, 1970) 551–565.

    Google Scholar 

  37. R.W. Cottle, J.S. Pang and V. Venkateswaran, “Sufficient matrices and the linear complementarity problem,”Linear Algebra and its Applications 114/115 (1989) 231–249.

    Google Scholar 

  38. R.W. Cottle and A.F. Veinott, Jr., “Polyhedral sets having a least element,”Mathematical Programming 3 (1972) 238–249.

    Google Scholar 

  39. S. Dafermos, “Traffic equilibria and variational inequalities,”Transportation Science 14 (1980) 42–54.

    Google Scholar 

  40. S. Dafermos, “The general multimodal network equilibrium problem with elastic demand,”Networks 12 (1982) 57–72.

    Google Scholar 

  41. S. Dafermos, “Relaxation algorithms for the general asymmetric traffic equilibrium problem,”Transportation Science 16 (1982) 231–240.

    Google Scholar 

  42. S. Dafermos, “An iterative scheme for variational inequalities,”Mathematical Programming 26 (1983) 40–47.

    Google Scholar 

  43. S. Dafermos, “Sensitivity analysis in variational inequalities,”Mathematics of Operations Research 13 (1988) 421–434.

    Google Scholar 

  44. S. Dafermos and A. Nagurney, “Sensitivity analysis for the general spatial economic equilibrium problem,”Operations Research 32 (1984) 1069–1086.

    Google Scholar 

  45. S. Dafermos and A. Nagurney, “Sensitivity analysis for the asymmetric network equilibrium problem,”Mathematical Programming 28 (1984) 174–184.

    Google Scholar 

  46. J.E. Dennis Jr. and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983).

    Google Scholar 

  47. I.C. Dolcetta and U. Mosco, “Implicit complementarity problems and quasi-variational inequalities,” in: R.W. Cottle, F. Giannessi and J.L. Lions, eds.,Variational Inequalities and Complementarity Problems: Theory and Applications (Wiley, New York, 1980) 75–87.

    Google Scholar 

  48. B.C. Eaves, “On the basic theorem of complementarity,”Mathematical Programming 1 (1971) 68–75.

    Google Scholar 

  49. B.C. Eaves, “The linear complementarity problem,”Management Science 17 (1971) 612–634.

    Google Scholar 

  50. B.C. Eaves, “Homotopies for computation of fixed points,”Mathematical Programming 3 (1972) 1–22.

    Google Scholar 

  51. B.C. Eaves, “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, RI, 1976) pp. 73–143.

    Google Scholar 

  52. B.C. Eaves, “Computing stationary points,”Mathematical Programming Study 7 (1978) 1–14.

    Google Scholar 

  53. B.C. Eaves, “Computing stationary points, again,” in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming 3 (Academic Press, New York, 1978) pp. 391–405.

    Google Scholar 

  54. B.C. Eaves, “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, 1983) pp. 63–78.

    Google Scholar 

  55. B.C. Eaves, “Thoughts on computing market equilibrium with SLCP,” Technical Report, Department of Operations Research, Stanford University (Stanford, CA, 1986).

    Google Scholar 

  56. S.C. Fang,Generalized Variational Inequality, Complementarity and Fixed Point Problems: Theory and Application. Ph.D. dissertation, Northwestern University (Evanston, IL, 1979).

    Google Scholar 

  57. S.C. Fang, “An iterative method for generalized complementarity problems,”IEEE Transactions on Automatic Control AC-25 (1980) 1225–1227.

    Google Scholar 

  58. S.C. Fang, “Traffic equilibria on multiclass user transportation networks analyzed via variational inequalities,”Tamkang Journal of Mathematics 13 (1982) 1–9.

    Google Scholar 

  59. S.C. Fang, “Fixed point models for the equilibrium problems on transportation networks,”Tamkang Journal of Mathematics 13 (1982) 181–191.

    Google Scholar 

  60. S.C. Fang, “A linearization method for generalized complementarity problems,”IEEE Transactions on Automatic Control AC 29 (1984) 930–933.

    Google Scholar 

  61. S.C. Fang and E.L. Peterson, “Generalized variational inequalities,”Journal of Optimization Theory and Application 38 (1982) 363–383.

    Google Scholar 

  62. S.C. Fang and E.L. Peterson, “General network equilibrium analysis,”International Journal of Systems Sciences 14 (1983) 1249–1257.

    Google Scholar 

  63. S.C. Fang and E.L. Peterson, “An economic equilibrium model on a multicommodity network,”International Journal of Systems Sciences 16 (1985) 479–490.

    Google Scholar 

  64. A.V. Fiacco,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic Press, New York, 1983).

    Google Scholar 

  65. A.V. Fiacco and J. Kyparisis, “Sensitivity analysis in nonlinear programming under second order assumptions,” in: A. Bagchi and H. Th. Jongen, eds.,Systems and Optimization (Springer, Berlin, 1985) pp. 74–97.

    Google Scholar 

  66. M. Fiedler and V. Ptak, “On matrices with nonpositive off-diagonal elements and positive principal minors,”Czechoslovak Mathematics Journal 12 (1962), 382–400.

    Google Scholar 

  67. M.L. Fisher and F.J. Gould, “A simplicial algorithm for the nonlinear complementarity problem,”Mathematical Programming 6 (1974) 281–300.

    Google Scholar 

  68. M.L. Fisher and J.W. Tolle, “The nonlinear complementarity problem: existence and determination of solutions,”SIAM Journal of Control and Optimization 15 (1977), 612–623.

    Google Scholar 

  69. C.S. Fisk and D.E. Boyce, “Alternative variational inequality formulations of the network equilibrium—travel choice problem,”Transportation Science 17 (1983) 454–463.

    Google Scholar 

  70. C.S. Fisk and S. Nguyen, “Solution algorithms for network equilibrium models with asymmetric user costs,”Transportation Science 16 (1982) 316–381.

    Google Scholar 

  71. M. Florian, ed.,Traffic Equilibrium Methods (Springer, Berlin, 1976).

    Google Scholar 

  72. M. Florian, “Nonlinear cost network models in transportation analysis,”Mathematical Programming Study 26 (1986) 167–196.

    Google Scholar 

  73. M. Florian and M. Los, “A new look at static spatial price equilibrium models,”Regional Science and Urban Economics 12 (1982) 579–597.

    Google Scholar 

  74. M. Florian and H. Spiess, “The convergence of diagonalization algorithms for asymmetric network equilibrium problems,”Transportation Research 16B (1982) 447–483.

    Google Scholar 

  75. T.L. Friesz, “Network equilibrium, design and aggregation,”Transportation Research 19A (1985) 413–427.

    Google Scholar 

  76. T.L. Friesz, R.L. Tobin, T.E. Smith and P.T. Harker, “A nonlinear complementary formulation and solution procedure for the general derived demand network equilibrium problem,”Journal of Regional Science 23 (1983) 337–359.

    Google Scholar 

  77. T.L. Friesz and P.T. Harker, “Freight network equilibrium: a review of the state of the art,” in: A. Daughety, ed.,Analytical Studies in Transportation Economics (Cambridge University Press, Cambridge, 1985) 161–206.

    Google Scholar 

  78. T.L. Friesz, P.T. Harker and R.L. Tobin, “Alternative algorithms for the general network spatial price equilibrium problem,”Journal of Regional Science 24 (1984) 473–507.

    Google Scholar 

  79. M. Fukushima, “A relaxed projection method for variational inequalities,”Mathematical Programming 35 (1986) 58–70.

    Google Scholar 

  80. D. Gabay and H. Moulin, “On the uniqueness and stability of Nash-equilibria in noncooperative games,” in: A. Bensoussan, P. Kleindorfer and C.S. Tapiero, eds.,Applied Stochastic Control in Econometrics and Management Science (North-Holland, Amsterdam, 1980) pp. 271–292.

    Google Scholar 

  81. C.B. Garcia and W.I. Zangwill,Pathways to Solutions, Fixed Points and Equilibria (Prentice-Hall, Englewood Cliffs, NJ, 1981).

    Google Scholar 

  82. R. Glowinski, J.L. Lions and R. Trémolières,Analyses Numérique des Inéquations Variationalles: Methodes Mathematiques de l'Informatique (Dunod, Paris, 1976).

    Google Scholar 

  83. C.D. Ha, “Application of degree theory in stability of the complementarity problem,”Mathematics of Operations Research 12 (1987) 368–376.

    Google Scholar 

  84. G.J. Habetler and M.M. Kostreva, “On a direct algorithm for nonlinear complementarity problems,”SIAM Journal of Control and Optimization 16 (1978) 504–511.

    Google Scholar 

  85. G.J. Habetler and A.L. Price, “Existence theory for generalized nonlinear complementarity problems,”Journal of Optimization Theory and Applications 7 (1971) 223–239.

    Google Scholar 

  86. J.H. Hammond,Solving Asymmetric Variational Inequality Problems and Systems of Equation with Generalized Nonlinear Programming Algorithms. Ph.D. dissertation, Department of Mathematics, M.I.T. (Cambridge, MA, 1984).

    Google Scholar 

  87. J.H. Hammond and T.L. Magnanti, “Generalized descent methods for asymmetric systems of equations,”Mathematics of Operations Research 12 (1987) 678–699.

    Google Scholar 

  88. J.H. Hammond and T.L. Magnanti, “A contracting ellipsoid method for variational inequality problems,” Working Paper OR 160-87, Operations Research Center, M.I.T. (Cambridge, MA, 1987).

    Google Scholar 

  89. T.H. Hansen,On the Approximation of a Competitive Equilibrium. Ph.D. dissertation, Department of Economics, Yale University (New Haven, CT, 1968).

    Google Scholar 

  90. T.H. Hansen and H. Scarf, “On the approximation of Nash equilibrium points in an N-person noncooperative game,”SIAM Journal of Applied Mathematics 26 (1974) 622–637.

    Google Scholar 

  91. P.T. Harker, “A variational inequality approach for the determination of oligopolistic market equilibrium,”Mathematical Programming 30 (1984) 105–111.

    Google Scholar 

  92. P.T. Harker, “A generalized spatial price equilibrium model,”Papers of the Regional Science Association 54 (1984) 25–42.

    Google Scholar 

  93. P.T. Harker, ed.,Spatial Price Equilibrium: Advances in Theory, Computation and Application. Lecture Notes in Economics and Mathematical Systems, Vol 249 (Springer, Berlin, 1985).

    Google Scholar 

  94. P.T. Harker, “Existence of competitive equilibria via Smith's nonlinear complementarity result,”Economics Letters 19 (1985) 1–4.

    Google Scholar 

  95. P.T. Harker, “Alternative models of spatial competition,”Operations Research 34 (1986) 410–425.

    Google Scholar 

  96. P.T. Harker, “A note on the existence of traffic equilibria,”Applied Mathematics and Computation 18 (1986) 277–283.

    Google Scholar 

  97. P.T. Harker,Predicting Intercity Freight Flows (VNU Science Press, Utrecht, The Netherlands, 1987).

    Google Scholar 

  98. P.T. Harker, “Multiple equilibria behaviors on networks,”Transportation Science 22 (1988), 39–46.

    Google Scholar 

  99. P.T. Harker, “Accelerating the convergence of the diagonalization and projection algorithms for finite-dimensional variational inequalities,”Mathematical Programming 41 (1988) 29–59.

    Google Scholar 

  100. P.T. Harker, “The core of a spatial price equilibrium game,”Journal of Regional Science 27 (1987) 369–389.

    Google Scholar 

  101. P.T. Harker, “Privatization of urban mass transportation: application of computable equilibrium models for network competition,”Transportation Science 22 (1988) 96–111.

    Google Scholar 

  102. P.T. Harker, “Generalized Nash games and quasivariational inequalities,” to appear in:European Journal of Operational Research.

  103. P.T. Harker and S.C. Choi, “A penalty function approach for mathematical programs with variational inequality constraints,” Working paper 87-09-08, Department of Decision Sciences, University of Pennsylvania (Philadelphia, PA, 1987).

    Google Scholar 

  104. P.T. Harker and J.S. Pang, “Existence of optimal solutions to mathematical programs with equilibrium constraints,”Operations Research Letters 7 (1988) 61–64.

    Google Scholar 

  105. P.T. Harker and J.S. Pang, “A damped-Newton method for the linear complementarity problem,” in: E.L. Allgower and K. Georg, eds.,Computational Solution of Nonlinear Systems of Equations. AMS Lectures on Applied Mathematics 26 (1990) 265–284.

    Google Scholar 

  106. P.T. Harker and J.S. Pang,Equilibrium Modeling With Variational Inequalities: Theory, Computation and Application, in preparation.

  107. P. Hartman and G. Stampacchia, “On some nonlinear elliptic differential functional equations,”Acta Mathematica 115 (1966) 153–188.

    Google Scholar 

  108. A. Haurie and P. Marcotte, “On the relationship between Nash-Cournot and Wardrop equilibria,”Networks 15 (1985) 295–308.

    Google Scholar 

  109. A. Haurie and P. Marcotte, “A game-theoretic approach to network equilibrium,”Mathematical Programming Study 26 (1986) 252–255.

    Google Scholar 

  110. A. Haurie, G. Zaccour, J. Legrand and Y. Smeers, “A stochastic dynamic Nash-Cournot model for the European gas market,” Working Paper G-87-24, École des Hautes Études Commeriales, Université de Montréal (Montréal, Que., 1987).

    Google Scholar 

  111. D.W. Hearn, “The gap function of a convex program,”Operations Research Letters 1 (1982) 67–71.

    Google Scholar 

  112. D.W. Hearn, S. Lawphongpanich and S. Nguyen, “Convex programming formulation of the asymmetric traffic assignment problem,”Transportation Research 18B (1984) 357–365.

    Google Scholar 

  113. D.W. Hearn, S. Lawphongpanich and J.A. Ventura, “Restricted simplicial decomposition: computation and extensions,”Mathematical Programming Study 31 (1987) 99–118.

    Google Scholar 

  114. W. Hildenbrand and A.P. Kirman,Introduction to Equilibrium Analysis (North-Holland, Amsterdam, 1976).

    Google Scholar 

  115. A.V. Holden, ed.,Chaos (Princeton University Press, Princeton, NJ, 1986).

    Google Scholar 

  116. T. Ichiishi,Game Theory for Economic Analysis (Academic Press, New York, 1983).

    Google Scholar 

  117. C.M. Ip,The Distorted Stationary Point Problem. Ph.D. dissertation, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, 1986).

    Google Scholar 

  118. K. Jittorntrum, “Solution point differentiability without strict complementarity in nonlinear programming,”Mathematical Programming Study 21 (1984) 127–138.

    Google Scholar 

  119. P.C. Jones, G. Morrison, J.C. Swarts and E. Theise, “Nonlinear spatial price equilibrium algorithms: a computational comparison,”Microcomputers in Civil Engineering 3 (1988) 265–271.

    Google Scholar 

  120. P.C. Jones, R. Saigal and M. Schneider, “Computing nonlinear network equilibria,”Mathematical Programming 31 (1985) 57–66.

    Google Scholar 

  121. N.H. Josephy, “Newton's method for generalized equations,” Technical Report No. 1965, Mathematics Research Center, University of Wisconsin (Madison, WI, 1979).

    Google Scholar 

  122. N.H. Josephy, “Quasi-Newton methods for generalized equations,” Technical Report No. 1966, Mathematics Research Center, University of Wisconsin (Madison, WI, 1979).

    Google Scholar 

  123. N.H. Josephy, “A Newton method for the PIES energy model,” Technical Summary Report No. 1977, Mathematics Research Center, University of Wisconsin (Madison, WI, 1979).

    Google Scholar 

  124. S. Karamardian, “The nonlinear complementarity problem with applications, parts I and II,”Journal of Optimization Theory and Applications 4 (1969) 87–98 and 167-81.

    Google Scholar 

  125. S. Karamardian, “Generalized complementarity problem,”Journal of Optimization Theory and Applications 8 (1971) 161–167.

    Google Scholar 

  126. S. Karamardian, “The complementarity problem,”Mathematical Programming 2 (1972) 107–129.

    Google Scholar 

  127. S. Karamardian, “Complementarity problems over cones with monotone and pseudomonotone maps,”Journal of Optimization Theory and Applications 18 (1976) 445–454.

    Google Scholar 

  128. S. Karamardian, “An existence theorem for the complementarity problem,”Journal of Optimization Theory and Applications 18 (1976) 445–454.

    Google Scholar 

  129. W. Karush,Minima of Functions of Several Variables with Inequalities as Side Conditions. M.S. thesis, Department of Mathematics, University of Chicago (Chicago, IL, 1939).

    Google Scholar 

  130. D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Application (Academic Press, New York, 1980).

    Google Scholar 

  131. M. Kojima, “Computational methods for solving the nonlinear complementarity problem,”Keio Engineering Reports 27 (1974) 1–41.

    Google Scholar 

  132. M. Kojima, “A unification of the existence theorems of the nonlinear complementarity problem,”Mathematical Programming 9 (1975) 257–277.

    Google Scholar 

  133. M. Kojima, “Strongly stable stationary solutions in nonlinear programming,” in: S.M. Robinson, ed.,Analysis and Computation of Fixed Points (Academic Press, New York, 1980) pp. 93–138.

    Google Scholar 

  134. M. Kojima, S. Mizuno, and T. Noma, “A new continuation method for complementarity problems with uniform P-functions,”Mathematical Programming 43 (1989) 107–114.

    Google Scholar 

  135. M. Kojima, S. Mizuno, and T. Noma, “Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems,” Research Report No. B-199, Department of Information Sciences, Tokyo Institute of Technology (Tokyo, Japan, 1988).

    Google Scholar 

  136. M.M. Kostreva, “Block pivot methods for solving the complementarity problem,”Linear Algebra and Its Application 21 (1978) 207–215.

    Google Scholar 

  137. M.M. Kostreva, “Elasto-hydrodynamic lubrication: a nonlinear complementarity problem,”International Journal for Numerical Methods in Fluids 4 (1984) 377–397.

    Google Scholar 

  138. H. Kremers and D. Talman, “Solving the nonlinear complementarity problem with lower and upper bounds,” FEW330, Department of Econometrics, Tilburg University (Tilburg, The Netherlands, 1988).

    Google Scholar 

  139. H.W. Kuhn, “Simplicial approximation of fixed points,”Proceedings of the National Academy of Sciences U.S.A. 61 (1968) 1238–1242.

    Google Scholar 

  140. H.W. Kuhn and A.W. Tucker, “Nonlinear programming,” in: J. Neyman, ed.,Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (University of California Press, Berkeley, CA, 1951) pp. 481–492.

    Google Scholar 

  141. J. Kyparisis, “Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems,”Mathematical Programming 36 (1986) 105–113.

    Google Scholar 

  142. J. Kyparisis, “Sensitivity analysis framework for variational inequalities,”Mathematical Programming 38 (1987) 203–213.

    Google Scholar 

  143. J. Kyparisis, “Perturbed solutions of variational inequality problems over polyhedral sets,”Journal of Optimization Theory and Applications 57 (1988) 295–305.

    Google Scholar 

  144. J. Kyparisis, “Sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers,” Working paper, Department of Decision Sciences and Information Systems, Florida International University (Miami, FL, 1987).

    Google Scholar 

  145. S. Lawphongpanich and D.W. Hearn, “Simplicial decomposition of asymmetric traffic assignment problem,”Transportation Research 18B (1984) 123–133.

    Google Scholar 

  146. S. Lawphongpanich and D.W. Hearn, “Bender's decomposition for variational inequalities,”Mathematical Programming (Series B) 48 (1990) 231–247, this issue.

    Google Scholar 

  147. S. Lawphongpanich and D.W. Hearn, “Restricted simplicial decomposition with application to the traffic assignment problem,”Ricera Operativa 38 (1986) 97–120.

    Google Scholar 

  148. L.J. LeBlanc, E.K. Morlok and W.P. Pierskalla, “An efficient approach to solving the road network equilibrium traffic assignment problem,”Transportation Research 9 (1974) 309–318.

    Google Scholar 

  149. C.E. Lemke, “Bimatrix equilibrium points and mathematical programming,”Management Science 11 (1965) 681–689.

    Google Scholar 

  150. C.E. Lemke and J.T. Howson, “Equilibrium points of bimatrix games,”SIAM Review 12 (1964) 45–78.

    Google Scholar 

  151. Y.Y. Lin and J.S. Pang, “Iterative methods for large convex quadratic programs: a survey,”SIAM Journal on Control and Optimization 25 (1987) 383–411.

    Google Scholar 

  152. J.L. Lions and G. Stampacchia, “Variational inequalities,”Communications on Pure and Applied Mathematics 20 (1967) 493–519.

    Google Scholar 

  153. H.J. Lüthi, “On the solution of variational inequality by the ellipsoid method,”Mathematics of Operations Research 10 (1985) 515–522.

    Google Scholar 

  154. T.L. Magnanti, “Models and algorithms for predicting urban traffic equilibria,” in: M. Florian, ed.,Transportation Planning Models (North-Holland, Amsterdam, 1984) pp. 153–185.

    Google Scholar 

  155. O. Mancino and G. Stampacchia, “Convex programming and variational inequalities,”Journal of Optimization Theory and Application 9 (1972) 3–23.

    Google Scholar 

  156. O.L. Mangasarian, “Equivalence of the complementarity problem to a system of nonlinear equations,”SIAM Journal on Applied Mathematics 31 (1976) 89–92.

    Google Scholar 

  157. O.L. Mangasarian, “Locally unique solutions of quadratic programs, linear and nonlinear complementarity problems,”Mathematical Programming 19 (1980) 200–212.

    Google Scholar 

  158. O.L. Mangasarian and L. McLinden, “Simple bounds for solutions of monotone complementarily problems and convex programs,”Mathematical Programming 32 (1985) 32–40.

    Google Scholar 

  159. A.S. Manne, “On the formulation and solution of economic equilibrium models,”Mathematical Programming Study 23 (1985) 1–22.

    Google Scholar 

  160. A.S. Manne and P.V. Preckel, “A three-region intertemporal model of energy, international trade and capital flows,”Mathematical Programming Study 23 (1985) 56–74.

    Google Scholar 

  161. P. Marcotte, “Network optimization with continuous control parameters,”Transportation Science 17 (1983) 181–197.

    Google Scholar 

  162. P. Marcotte, “Quelques notes et résultats nouveaux sur les problème d'equilibre d'un oligopole,”R.A.I.R.O. Recherche Opérationnelle 18 (1984) 147–171.

    Google Scholar 

  163. P. Marcotte, “A new algorithm for solving variational inequalities with application to the traffic assignment problem,”Mathematical Programming 33 (1985) 339–351.

    Google Scholar 

  164. P. Marcotte, “Gap-decreasing algorithms for monotone variational inequalities,” paper presented at the ORSA/TIMS Meeting, Miami Beach, October 1986.

  165. P. Marcotte, “Network design with congestion effects: a case of bi-level programming,”Mathematical Programming 34 (1986) 142–162.

    Google Scholar 

  166. P. Marcotte and J.P. Dussault, “A modified Newton method for solving variational inequalities,”Proceeding of the 24th IEEE Conference on Decision and Control, pp. 1433–1436.

  167. P. Marcotte and J.P. Dussault, “A note on a globally convergent Newton method for solving monotone variational inequalities,”Operations Research Letters 6 (1987) 35–42.

    Google Scholar 

  168. L. Mathiesen, “Computation of economic equilibria by a sequence of linear complementarity problems,”Mathematical Programming Study 23 (1985) 144–162.

    Google Scholar 

  169. L. Mathiesen, “Computational experience in solving equilibrium models by a sequence of linear complementarity problems,”Operations Research 33 (1985) 1225–1250.

    Google Scholar 

  170. L. Mathiesen, “An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example,”Mathematical Programming 37 (1987) 1–18.

    Google Scholar 

  171. L. Mathiesen and A. Lont, “Modeling market equilibria: an application to the world steel market,” Working Paper MU04, Center for Applied Research, Norwegian School of Economics and Business Administration (Bergen, Norway, 1983).

    Google Scholar 

  172. L. Mathiesen and E. Steigum, Jr., “Computation of unemployment equilibria in a two-country multi-period model with neutral money,” Working Paper, Center for Applied Research, Norwegian School of Economics and Business Administration (Bergen, Norway, 1985).

    Google Scholar 

  173. L. McKenzie, “Why compute economic equilibria?,” in:Computing Equilibria: How and Why (North-Holland, Amsterdam, 1976).

    Google Scholar 

  174. L. McLinden, “The complementarity problem for maximal monotone multifunctions,” in: R.W. Cottle, F. Giannessi and J.L. Lions, eds.,Variational Inequalities and Complementarity Problems (Academic Press, New York, 1980) pp. 251–270.

    Google Scholar 

  175. L. McLinden, “An analogue of Moreau's proximation theorem, with application to the nonlinear complementarity problem,”Pacific Journal of Mathematics 88 (1980) 101–161.

    Google Scholar 

  176. L. McLinden, “Stable monotone variational inequalities,”Mathematical Programming (Series B) 48 (1990) 303–338, this issue.

    Google Scholar 

  177. N. Megiddo, “A monotone complementarity problem with feasible solutions but no complementary solutions,”Mathematical Programming 12 (1977) 131–132.

    Google Scholar 

  178. N. Megiddo, “On the parametric nonlinear complementarity problem,”Mathematical Programming Study 7 (1978) 142–159.

    Google Scholar 

  179. N. Megiddo and M. Kojima, “On the existence and uniqueness of solutions in nonlinear complementarity theory,”Mathematical Programming 12 (1977) 110–130.

    Google Scholar 

  180. G.J. Minty, “Monotone (non-linear) operators in Hilbert space,”Duke Mathematics Journal 29 (1962) 341–346.

    Google Scholar 

  181. J.J. Moré, “The application of variational inequalities to complementarity problems and existence theorems,” Technical Report 71–90, Department of Computer Sciences, Cornell University (Ithaca, NY, 1971).

    Google Scholar 

  182. J.J. Moré, “Classes of functions and feasibility conditions in nonlinear complementarity problems,”Mathematical Programming 6 (1974) 327–338.

    Google Scholar 

  183. J.J. Moré, “Coercivity conditions in nonlinear complementarity problems,”SIAM Review 17 (1974) 1–16.

    Google Scholar 

  184. J.J. Moré and W.C. Rheinboldt, “On P- and S-functions and related classes of n-dimensional nonlinear mappings,”Linear Algebra and Its Applications 6 (1973) 45–68.

    Google Scholar 

  185. J.J. Moreau, “Proximitè et dualitè dans un espace Hilberiten,”Bulletin of the Society of Mathematics of France 93 (1965) 273–299.

    Google Scholar 

  186. J.D. Murchland, “Braess' paradox of traffic flow,”Transportation Research 4 (1970) 391–394.

    Google Scholar 

  187. K.G. Murty,Linear Complementarity, Linear and Nonlinear Programming (Helderman, Berlin, 1988).

    Google Scholar 

  188. A. Nagurney, “Comparative tests of multimodal traffic equilibrium methods,”Transportation Research 18B (1984) 469–485.

    Google Scholar 

  189. A. Nagurney, “Computational comparisons of algorithms for general asymmetric traffic equilibrium problems with fixed and elastic demand,”Transportation Research 20B (1986) 78–84.

    Google Scholar 

  190. A. Nagurney, “Computational comparisons of spatial price equilibrium methods,”Journal of Regional Science 27 (1987) 55–76.

    Google Scholar 

  191. A. Nagurney, “Competitive equilibrium problems, variational inequalities and regional science,”Journal of Regional Science 27 (1987) 503–517.

    Google Scholar 

  192. J.F. Nash, “Equilibrium points in n-person games,”Proceedings of the National Academy of Sciences 36 (1950) 48–49.

    Google Scholar 

  193. S. Nguyen and C. Dupuis, “An efficient method for computing traffic equilibria in networks with asymmetric transportation costs,”Transportation Science 18 (1984) 185–202.

    Google Scholar 

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

    Google Scholar 

  195. A.R. Pagan and J.H. Shannon, “Sensitivity analysis for linearized computable general equilibrium models,” in: J. Piggott and J. Whalley, eds.,New Developments in Applied General Equilibrium Analysis (Cambridge University Press, Cambridge, 1985) pp. 104–118.

    Google Scholar 

  196. J.S. Pang,Least-Element Complementarity Theory. Ph.D. dissertation, Department of Operations Research, Stanford University (Stanford, CA, 1976).

    Google Scholar 

  197. J.S. Pang, “The implicit complementarity problem“, in: O.L. Managasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming 4 (Academic Press, New York, 1981) 487–518.

    Google Scholar 

  198. J.S. Pang, “A column generation technique for the computation of stationary points,”Mathematics of Operations Research 6 (1981) 213–244.

    Google Scholar 

  199. J.S. Pang, “On the convergence of a basic iterative method for the implicit complementarity problem,”Journal of Optimization Theory and Application 37 (1982) 149–162.

    Google Scholar 

  200. J.S. Pang, “Solution of the general multicommodity spatial equilibrium problem by variational and complementarity methods,”Journal of Regional Science 24 (1984) 403–414.

    Google Scholar 

  201. J.S. Pang, “Variational inequality problems over product sets: applications and iterative methods,”Mathematical Programming 31 (1985) 206–219.

    Google Scholar 

  202. J.S. Pang, “Inexact Newton methods for the nonlinear complementarity problem,”Mathematical Programming 36 (1986) 54–71.

    Google Scholar 

  203. J.S. Pang, “A posteriori error bounds for linearly constrained variational inequality problems,”Mathematics of Operations Research 12 (1987) 474–484.

    Google Scholar 

  204. J.S. Pang, “Two characterization theorems in complementarity theory,”Operations Research Letters 7 (1988) 27–31.

    Google Scholar 

  205. J.S. Pang, “Newton's method for B-differentiable equations,” to appear in:Mathematics of Operations Research.

  206. J.S. Pang, “Solution differentiability and continuation of Newton's method for variational inequality problems over polyhedral sets,” to appear in:Journal of Optimization Theory and Applications.

  207. J.S. Pang and D. Chan, “Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313.

    Google Scholar 

  208. J.S. Pang and J.M. Yang, “Parallel Newton methods for the nonlinear complementarity problem,”Mathematical Programming (Series B) 42 (1988) 407–420.

    Google Scholar 

  209. J.S. Pang and C.S. Yu, “Linearized simplicial decomposition methods for computing traffic equilibria on networks,”Networks 14 (1984) 427–438.

    Google Scholar 

  210. P.V. Preckel, “Alternative algorithms for computing economic equilibria,”Mathematical Programming Study 23 (1985) 163–172.

    Google Scholar 

  211. P.V. Preckel, “A modified Newton method for the nonlinear complementarity problem and its implementation,” paper presented at the ORSA/TIMS Meeting, Miami Beach, FL, October 1986.

  212. Y. Qiu and T.L. Magnanti, “Sensitivity analysis for variational inequalities defined on polyhedral sets,”Mathematics of Operations Research 14 (1989) 410–432.

    Google Scholar 

  213. Y. Qiu and T.L. Magnanti, “Sensitivity analysis for variational inequalities,” Working Paper OR 163-87, Operations Research Center, M.I.T. (Cambridge, MA, 1987).

    Google Scholar 

  214. A. Reinoza,A Degree For Generalized Equations. Ph.D. dissertation, Department of Industrial Engineering, University of Wisconsin (Madison, WI, 1979).

    Google Scholar 

  215. A. Reinoza, “The strong positivity conditions,”Mathematics of Operations Research 10 (1985) 54–62.

    Google Scholar 

  216. W.C. Rheinboldt,Numerical Analysis of Parameterized Nonlinear Equations (Wiley, New York, 1986).

    Google Scholar 

  217. S.M. Robinson, “Generalized equations and their solutions, part I: basic theory,”Mathematical Programming Study 10 (1979) 128–141.

    Google Scholar 

  218. S.M. Robinson, “Strongly regular generalized equations,”Mathematics of Operations Research 5 (1980) 43–62.

    Google Scholar 

  219. S.M. Robinson, “Generalized equations and their solutions, part II: applications to nonlinear programming,”Mathematical Programming Study 19 (1982) 200–221.

    Google Scholar 

  220. S.M. Robinson, “Generalized equations,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming: The State of the Art (Springer, Berlin, 1982) pp. 346–367.

    Google Scholar 

  221. S.M. Robinson, “Implicit B-differentiability in generalized equations,” Technical Summary Report No. 2854, Mathematics Research Center, University of Wisconsin (Madison, WI, 1985).

    Google Scholar 

  222. S.M. Robinson, “Local structure of feasible sets in nonlinear programming, part III: stability and sensitivity,”Mathematical Programming Study 30 (1987) 45–66.

    Google Scholar 

  223. S.M. Robinson, “An implicit-function theorem for a class of nonsmooth functions,” to appear in:Mathematics of Operations Research.

  224. R.T. Rockafellar, “Characterization of the subdifferentials of convex functions,”Pacific Journal of Mathematics 17 (1966) 497–510.

    Google Scholar 

  225. R.T. Rockafellar, “Convex functions, monotone operators, and variational inequalities,”Theory and Applications of Monotone Operators: Proceedings of the NATO Advanced Study Institute, Venice, Italy (Edizioni Oderisi, Gubbio, Italy, 1968) pp. 35–65.

    Google Scholar 

  226. R.T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,”Pacific Journal of Mathematics 33 (1970) 209–216.

    Google Scholar 

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

    Google Scholar 

  228. R.T. Rockafellar, “Augmented Lagrangian and application of the proximal point algorithm in convex programming,”Mathematics of Operations Research 1 (1976) 97–116.

    Google Scholar 

  229. R.T. Rockafellar, “Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization 14 (1976) 877–898.

    Google Scholar 

  230. R.T. Rockafellar, “Lagrange multipliers and variational inequalities,” in: R.W. Cottle, F. Giannessi, and J.L.Lions, eds.,Variational Inequalities and Complementarity Problems: Theory and Applications (Wiley, New York, 1980) pp. 303–322.

    Google Scholar 

  231. T.F. Rutherford,Applied General Equilibrium Modeling. Ph.D. dissertation Department of Operations Research, Stanford University (Stanford, CA, 1986).

    Google Scholar 

  232. T.F. Rutherford, “Implementation issues and computational performance solving applied general equilibrium models with SLCP,” Discussion Paper 837, Cowles Foundation for Research in Economics, Yale University (New Haven, CT, 1987).

    Google Scholar 

  233. R. Saigal, “Extension of the generalized complementarity problem,”Mathematics of Operations Research 1 (1976) 260–266.

    Google Scholar 

  234. P.A. Samuelson, “Spatial price equilibrium and linear programming,”American Economic Review 42 (1952) 283–303.

    Google Scholar 

  235. H.E. Scarf, “The approximation of fixed points of a continuous mapping,”SIAM Journal on Applied Mathematics 15 (1967) 1328–1342.

    Google Scholar 

  236. H.E. Scarf and T. Hansen,Computation of Economic Equilibria (Yale University Press, New Haven, CT, 1973).

    Google Scholar 

  237. A. Shapiro, “On concepts of directional differentiability,” Research Report 73/88(18), Department of Mathematics and Applied Mathematics, University of South Africa (Pretoria, South Africa, 1988).

    Google Scholar 

  238. J.B. Shoven, “Applying fixed points algorithms to the analysis of tax policies,” in: S. Karmardian and C.B. Garcia, eds.,Fixed Points: Algorithms and Applications (Academic Press, New York, 1977) pp. 403–434.

    Google Scholar 

  239. J.B. Shoven, “The application of fixed point methods to economics,” in: B.C. Eaves, F.J. Gould, H.O. Peitgen, and M.J. Todd, eds.,Homotopy Methods and Global Convergence (Plenum Press, New York, 1983) pp. 249–262.

    Google Scholar 

  240. S. Smale, “A convergent process of price adjustment and global Newton methods,”Journal of Mathematical Economics 3 (1976) 107–120.

    Google Scholar 

  241. M.J. Smith, “The existence, uniqueness and stability of traffic equilibria,”Transportation Research 13B (1979) 295–304.

    Google Scholar 

  242. M.J. Smith, “The existence and calculation of traffic equilibria,”Transportation Research 17B (1983) 291–303.

    Google Scholar 

  243. M.J. Smith, “A descent algorithm for solving monotone variational inequality and monotone complementarity problems,”Journal of Optimization Theory and Application 44 (1984) 485–496.

    Google Scholar 

  244. M.J. Smith, “The stability of a dynamic model of traffic assignment- an application of a method of Lyapunov,”Transportation Science 18 (1984) 245–252.

    Google Scholar 

  245. T.E. Smith, “A solution condition for complementarity problems: with an apilication to spatial price equilibrium,”Applied Mathematics and Computation 15 (1984) 61–69.

    Google Scholar 

  246. J.E. Spingarn, “Partial inverse of a monotone operator,”Applied Mathematics and Optimization 10 (1983) 247–265.

    Google Scholar 

  247. J.E. Spingarn, “Applications of the method of partial inverses to convex programming: decomposition,”Mathematical Programming 32 (1985) 199–223.

    Google Scholar 

  248. J.E. Spingarn, “On computation of spatial economic equilibria,” Discussion Paper 8731, Center for Operations Research and Econometrics, Université Catholique de Louvain (Louvain-la-Neuve, Belgium, 1987).

    Google Scholar 

  249. G. Stampacchia, “Variational inequalities,” inTheory and Applications of Monotone Operators, Proceedings of the NATO Advanced Study Institute, Venice, Italy (Edizioni Oderisi, Gubbio, Italy, 1968) pp. 102–192.

    Google Scholar 

  250. R. Steinberg and R.E. Stone, “The prevalence of paradoxes in transportation equilibrium problems,” Working paper, AT&T Bell Laboratories (Holmdel, NJ, 1987).

    Google Scholar 

  251. R. Steinberg and W.I. Zangwill, “The prevalence of Braess' paradox,”Transportation Science 17 (1983) 301–319.

    Google Scholar 

  252. J.C. Stone, “Sequential optimization and complementarity techniques for computing economic equilibria,”Mathematical Programming Study 23 (1985) 173–191.

    Google Scholar 

  253. P.K. Subramanian, “Gauss-Newton methods for the nonlinear complementarity problem,” Technical Summary Report No. 2845, Mathematics Research Center, University of Wisconsin (Madison, WI, 1985).

    Google Scholar 

  254. P.K. Subramanian, “Fixed-point methods for the complementarity problem,” Technical Summary Report No. 2857, Mathematics Research Center, University of Wisconsin (Madison, WI, 1985).

    Google Scholar 

  255. P.K. Subramanian, “A note on least two norm solutions of monotone complementarity problems,”Applied Mathematics Letters 1 (1988) 395–397.

    Google Scholar 

  256. A. Tamir, “Minimality and complementarity properties associated with Z-functions and Mfunctions,”Mathematical Programming 7 (1974) 17–31.

    Google Scholar 

  257. R.L. Tobin, “General spatial price equilibria: sensitivity analysis for variational inequality and nonlinear complementarity formulations,” in: P.T. Harker, ed.,Spatial Price Equilibrium: Advances in Theory, Computation and Application, Lecture Notes in Economics and Mathematical Systems, Vol. 249 (Springer, Berlin, 1985) pp. 158–195.

    Google Scholar 

  258. R.L. Tobin, “Sensitivity analysis for variational inequalities,”Journal of Optimization Theory and Applications 48 (1986) 191–204.

    Google Scholar 

  259. M.J. Todd,The Computation of Fixed Points and Applications (Springer, Berlin, 1976).

    Google Scholar 

  260. M.J. Todd, “A note on computing equilibria in economics with activity models of production“,Journal of Mathematical Economics 6 (1979) 135–144.

    Google Scholar 

  261. G. Van der Laan and A.J.J. Talman, “Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds,”Mathematical Programming 38 (1987) 1–15.

    Google Scholar 

  262. J.A. Ventura and D.W. Hearn, “Restricted simplicial decomposition for convex constrained problems,” Research Report No. 86-15, Department of Industrial and Systems Engineering, University of Florida (Gainesville, FL, 1986).

    Google Scholar 

  263. J.G. Wardrop, “Some theoretical aspects of road traffic research,”Proceedings of the Institute of Civil Engineers, Part II (1952) 325–378.

  264. L.T. Watson, “Solving the nonlinear complementarity problem by a homotopy method,”SIAM Journal on Control and Optimization 17 (1979) 36–46.

    Google Scholar 

  265. J. Whalley, “Fiscal harmonization in the EEC: some preliminary findings of fixed point calculations,” in: S. Karamardian and C.B. Garcia, eds.,Fixed Points: Algorithms and Applications (Academic Press, New York, 1977) pp. 435–472.

    Google Scholar 

  266. Y. Yamamoto, “A path following algorithm for stationary point problems,”Journal of the Operations Research Society of Japan 30 (1987) 181–198.

    Google Scholar 

  267. Y. Yamamoto, “Fixed point algorithms for stationary point problems,” in: M. Zri and K. Tanabe, eds.,Mathematical Programming: Recent Developments and Applications (KTK Scientific Publishers, Tokyo, 1989) pp. 283–308.

    Google Scholar 

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The research of this author was supported by the National Science Foundation Presidential Young Investigator Award ECE-8552773 and by the AT&T Program in Telecommunications Technology at the University of Pennsylvania.

The research of this author was supported by the National Science Foundation under grant ECS-8644098.

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Harker, P.T., Pang, JS. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Mathematical Programming 48, 161–220 (1990). https://doi.org/10.1007/BF01582255

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