Find out how to access previewonly content
Finitedimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications
 Patrick T. Harker,
 JongShi Pang
 … show all 2 hide
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Abstract
Over the past decade, the field of finitedimensional 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, socioeconomic analysis, energy modeling, and game theory. This paper provides a stateoftheart review of these developments as well as a summary of some open research topics in this growing field.
The research of this author was supported by the National Science Foundation Presidential Young Investigator Award ECE8552773 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 ECS8644098.
 H.Z. Aashtiani and T.L. Magnanti, “Equilibria on a congested transportation network,”SIAM Journal on Algebraic and Discrete Methods 2 (1981) 213–226.
 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.
 M. Abdulaal and L.J. LeBlanc, “Continuous equilibrium network design models,”Transportation Research 13B (1979) 19–32.
 M. Aganagic, “Variational inequalities and generalized complementarity problems,” Technical Report SOL 7811, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA 1978).
 B.H. Ahn,Computation of Market Equilibria for Policy Analysis: The Project Independence Evaluation Study (PIES) Approach (Garland, NY, 1979).
 B.H. Ahn, “A GaussSeidel iteration method for nonlinear variational inequality problems over rectangles,”Operations Research Letters 1 (1982) 117–120.
 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).
 B.H. Ahn and W.W. Hogan, “On convergence of the PIES algorithm for computing equilibria,”Operations Research 30 (1982) 281–300.
 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.
 R. Asmuth, “Traffic network equilibrium,” Technical Report SOL 782, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1978).
 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.
 J.P. Aubin,Mathematical Methods of Game and Economic Theory (NorthHolland, Amsterdam, 1979).
 M. Avriel,Nonlinear Programming: Analysis and Methods (PrenticeHall, Englewood Cliffs, NJ, 1976).
 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.
 C. Baiocchi and A. Capelo,Variational and Quasivariational Inequalities: Application to FreeBoundary Problems (Wiley, New York, 1984).
 B. Banks, J. Guddat, D. Klatte, B. Kummer and K. Tammer,Nonlinear Parametric Optimization (Birkhauser, Basel, 1983).
 V. Barbu,Optimal Control of Variational Inequalities (Pitman Advanced Publishing Program, Boston, 1984).
 M.J. Beckman, C.B. McGuire, and C.B. Winston,Studies in the Economics of Transportation (Yale University Press, New Haven, CT, 1956).
 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.
 A. Bensoussan, M. Goursat and J.L. Lions, “Contrôle impulsionnel et inéquations quasivariationnelles stationnaires,”Comptes Rendus Academie Sciences Paris 276 (1973) 1279–1284.
 A. Bensoussan and J.L. Lions, “Nouvelle formulation de problèmes de contrôle impulsionnel et applications,”Comptes Rendus Academie Sciences Paris 276 (1973) 1189–1192.
 A. Bensoussan and J.L. Lions, “Nouvelles méthodes en contrôle impulsionnel,”Applied Mathematics and Optimization 1 (1974) 289–312.
 C. Berge,Topological Spaces (Oliver and Boyd, Edinburgh, Scotland, 1963).
 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.
 K.C. Border,Fixed Point Theorems with Applications to Economics and Game Theory (Cambridge University Press, Cambridge, 1985).
 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.
 M. Carey, “Integrability and mathematical programming models: a survey and parametric approach,”Econometrica 45 (1977) 1957–1976.
 D. Chan and J.S. Pang, “The generalized quasivariational inequality problem,”Mathematics of Operations Research 7 (1982) 211–222.
 G.S. Chao and T.L. Friesz, “Spatial price equilibrium sensitivity analysis,”Transportation Research 18B (1984) 423–440.
 S.C. Choi, W.S. DeSarbo and P.T. Harker, “Product positioning under price competition,”Management Science 36 (1990) 265–284.
 R.W. Cottle,Nonlinear Programs with Positively Bounded Jacobians. Ph.D. dissertation, Department of Mathematics, University of California (Berkeley, CA, 1964).
 R.W. Cottle, “Nonlinear programs with positively bounded Jacobians,”SIAM Journal on Applied Mathematics 14 (1966) 147–158.
 R.W. Cottle, “Complementarity and variational problems,”Symposia Mathematica XIX (1976) 177–208.
 R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming,”Linear Algebra and Its Applications 1 (1968) 103–125.
 R.W. Cottle, F. Giannessi and J.L. Lions, eds.,Variational Inequalities and Complementarity Problems: Theory and Applications (Wiley, New York, 1980).
 R.W. Cottle, G.J. Habetler and C.E. Lemke, “Quadratic forms semidefinite over convex cones,” in: H.W. Kuhn, ed.,Proceedings of the Princeton Symposium on Mathematical Programming (Princeton University Press, Princeton, NJ, 1970) 551–565.
 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.
 R.W. Cottle and A.F. Veinott, Jr., “Polyhedral sets having a least element,”Mathematical Programming 3 (1972) 238–249.
 S. Dafermos, “Traffic equilibria and variational inequalities,”Transportation Science 14 (1980) 42–54.
 S. Dafermos, “The general multimodal network equilibrium problem with elastic demand,”Networks 12 (1982) 57–72.
 S. Dafermos, “Relaxation algorithms for the general asymmetric traffic equilibrium problem,”Transportation Science 16 (1982) 231–240.
 S. Dafermos, “An iterative scheme for variational inequalities,”Mathematical Programming 26 (1983) 40–47.
 S. Dafermos, “Sensitivity analysis in variational inequalities,”Mathematics of Operations Research 13 (1988) 421–434.
 S. Dafermos and A. Nagurney, “Sensitivity analysis for the general spatial economic equilibrium problem,”Operations Research 32 (1984) 1069–1086.
 S. Dafermos and A. Nagurney, “Sensitivity analysis for the asymmetric network equilibrium problem,”Mathematical Programming 28 (1984) 174–184.
 J.E. Dennis Jr. and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (PrenticeHall, Englewood Cliffs, NJ, 1983).
 I.C. Dolcetta and U. Mosco, “Implicit complementarity problems and quasivariational 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.
 B.C. Eaves, “On the basic theorem of complementarity,”Mathematical Programming 1 (1971) 68–75.
 B.C. Eaves, “The linear complementarity problem,”Management Science 17 (1971) 612–634.
 B.C. Eaves, “Homotopies for computation of fixed points,”Mathematical Programming 3 (1972) 1–22.
 B.C. Eaves, “A short course in solving equations with PL homotopies,” in: R.W. Cottle and C.E. Lemke eds.,Nonlinear Programming: SIAMAMS Proceedings 9 (American Mathematical Society, Providence, RI, 1976) pp. 73–143.
 B.C. Eaves, “Computing stationary points,”Mathematical Programming Study 7 (1978) 1–14.
 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.
 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.
 B.C. Eaves, “Thoughts on computing market equilibrium with SLCP,” Technical Report, Department of Operations Research, Stanford University (Stanford, CA, 1986).
 S.C. Fang,Generalized Variational Inequality, Complementarity and Fixed Point Problems: Theory and Application. Ph.D. dissertation, Northwestern University (Evanston, IL, 1979).
 S.C. Fang, “An iterative method for generalized complementarity problems,”IEEE Transactions on Automatic Control AC25 (1980) 1225–1227.
 S.C. Fang, “Traffic equilibria on multiclass user transportation networks analyzed via variational inequalities,”Tamkang Journal of Mathematics 13 (1982) 1–9.
 S.C. Fang, “Fixed point models for the equilibrium problems on transportation networks,”Tamkang Journal of Mathematics 13 (1982) 181–191.
 S.C. Fang, “A linearization method for generalized complementarity problems,”IEEE Transactions on Automatic Control AC 29 (1984) 930–933.
 S.C. Fang and E.L. Peterson, “Generalized variational inequalities,”Journal of Optimization Theory and Application 38 (1982) 363–383.
 S.C. Fang and E.L. Peterson, “General network equilibrium analysis,”International Journal of Systems Sciences 14 (1983) 1249–1257.
 S.C. Fang and E.L. Peterson, “An economic equilibrium model on a multicommodity network,”International Journal of Systems Sciences 16 (1985) 479–490.
 A.V. Fiacco,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic Press, New York, 1983).
 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.
 M. Fiedler and V. Ptak, “On matrices with nonpositive offdiagonal elements and positive principal minors,”Czechoslovak Mathematics Journal 12 (1962), 382–400.
 M.L. Fisher and F.J. Gould, “A simplicial algorithm for the nonlinear complementarity problem,”Mathematical Programming 6 (1974) 281–300.
 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.
 C.S. Fisk and D.E. Boyce, “Alternative variational inequality formulations of the network equilibrium—travel choice problem,”Transportation Science 17 (1983) 454–463.
 C.S. Fisk and S. Nguyen, “Solution algorithms for network equilibrium models with asymmetric user costs,”Transportation Science 16 (1982) 316–381.
 M. Florian, ed.,Traffic Equilibrium Methods (Springer, Berlin, 1976).
 M. Florian, “Nonlinear cost network models in transportation analysis,”Mathematical Programming Study 26 (1986) 167–196.
 M. Florian and M. Los, “A new look at static spatial price equilibrium models,”Regional Science and Urban Economics 12 (1982) 579–597.
 M. Florian and H. Spiess, “The convergence of diagonalization algorithms for asymmetric network equilibrium problems,”Transportation Research 16B (1982) 447–483.
 T.L. Friesz, “Network equilibrium, design and aggregation,”Transportation Research 19A (1985) 413–427.
 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.
 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.
 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.
 M. Fukushima, “A relaxed projection method for variational inequalities,”Mathematical Programming 35 (1986) 58–70.
 D. Gabay and H. Moulin, “On the uniqueness and stability of Nashequilibria in noncooperative games,” in: A. Bensoussan, P. Kleindorfer and C.S. Tapiero, eds.,Applied Stochastic Control in Econometrics and Management Science (NorthHolland, Amsterdam, 1980) pp. 271–292.
 C.B. Garcia and W.I. Zangwill,Pathways to Solutions, Fixed Points and Equilibria (PrenticeHall, Englewood Cliffs, NJ, 1981).
 R. Glowinski, J.L. Lions and R. Trémolières,Analyses Numérique des Inéquations Variationalles: Methodes Mathematiques de l'Informatique (Dunod, Paris, 1976).
 C.D. Ha, “Application of degree theory in stability of the complementarity problem,”Mathematics of Operations Research 12 (1987) 368–376.
 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.
 G.J. Habetler and A.L. Price, “Existence theory for generalized nonlinear complementarity problems,”Journal of Optimization Theory and Applications 7 (1971) 223–239.
 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).
 J.H. Hammond and T.L. Magnanti, “Generalized descent methods for asymmetric systems of equations,”Mathematics of Operations Research 12 (1987) 678–699.
 J.H. Hammond and T.L. Magnanti, “A contracting ellipsoid method for variational inequality problems,” Working Paper OR 16087, Operations Research Center, M.I.T. (Cambridge, MA, 1987).
 T.H. Hansen,On the Approximation of a Competitive Equilibrium. Ph.D. dissertation, Department of Economics, Yale University (New Haven, CT, 1968).
 T.H. Hansen and H. Scarf, “On the approximation of Nash equilibrium points in an Nperson noncooperative game,”SIAM Journal of Applied Mathematics 26 (1974) 622–637.
 P.T. Harker, “A variational inequality approach for the determination of oligopolistic market equilibrium,”Mathematical Programming 30 (1984) 105–111.
 P.T. Harker, “A generalized spatial price equilibrium model,”Papers of the Regional Science Association 54 (1984) 25–42.
 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).
 P.T. Harker, “Existence of competitive equilibria via Smith's nonlinear complementarity result,”Economics Letters 19 (1985) 1–4.
 P.T. Harker, “Alternative models of spatial competition,”Operations Research 34 (1986) 410–425.
 P.T. Harker, “A note on the existence of traffic equilibria,”Applied Mathematics and Computation 18 (1986) 277–283.
 P.T. Harker,Predicting Intercity Freight Flows (VNU Science Press, Utrecht, The Netherlands, 1987).
 P.T. Harker, “Multiple equilibria behaviors on networks,”Transportation Science 22 (1988), 39–46.
 P.T. Harker, “Accelerating the convergence of the diagonalization and projection algorithms for finitedimensional variational inequalities,”Mathematical Programming 41 (1988) 29–59.
 P.T. Harker, “The core of a spatial price equilibrium game,”Journal of Regional Science 27 (1987) 369–389.
 P.T. Harker, “Privatization of urban mass transportation: application of computable equilibrium models for network competition,”Transportation Science 22 (1988) 96–111.
 P.T. Harker, “Generalized Nash games and quasivariational inequalities,” to appear in:European Journal of Operational Research.
 P.T. Harker and S.C. Choi, “A penalty function approach for mathematical programs with variational inequality constraints,” Working paper 870908, Department of Decision Sciences, University of Pennsylvania (Philadelphia, PA, 1987).
 P.T. Harker and J.S. Pang, “Existence of optimal solutions to mathematical programs with equilibrium constraints,”Operations Research Letters 7 (1988) 61–64.
 P.T. Harker and J.S. Pang, “A dampedNewton 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.
 P.T. Harker and J.S. Pang,Equilibrium Modeling With Variational Inequalities: Theory, Computation and Application, in preparation.
 P. Hartman and G. Stampacchia, “On some nonlinear elliptic differential functional equations,”Acta Mathematica 115 (1966) 153–188.
 A. Haurie and P. Marcotte, “On the relationship between NashCournot and Wardrop equilibria,”Networks 15 (1985) 295–308.
 A. Haurie and P. Marcotte, “A gametheoretic approach to network equilibrium,”Mathematical Programming Study 26 (1986) 252–255.
 A. Haurie, G. Zaccour, J. Legrand and Y. Smeers, “A stochastic dynamic NashCournot model for the European gas market,” Working Paper G8724, École des Hautes Études Commeriales, Université de Montréal (Montréal, Que., 1987).
 D.W. Hearn, “The gap function of a convex program,”Operations Research Letters 1 (1982) 67–71.
 D.W. Hearn, S. Lawphongpanich and S. Nguyen, “Convex programming formulation of the asymmetric traffic assignment problem,”Transportation Research 18B (1984) 357–365.
 D.W. Hearn, S. Lawphongpanich and J.A. Ventura, “Restricted simplicial decomposition: computation and extensions,”Mathematical Programming Study 31 (1987) 99–118.
 W. Hildenbrand and A.P. Kirman,Introduction to Equilibrium Analysis (NorthHolland, Amsterdam, 1976).
 A.V. Holden, ed.,Chaos (Princeton University Press, Princeton, NJ, 1986).
 T. Ichiishi,Game Theory for Economic Analysis (Academic Press, New York, 1983).
 C.M. Ip,The Distorted Stationary Point Problem. Ph.D. dissertation, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, 1986).
 K. Jittorntrum, “Solution point differentiability without strict complementarity in nonlinear programming,”Mathematical Programming Study 21 (1984) 127–138.
 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.
 P.C. Jones, R. Saigal and M. Schneider, “Computing nonlinear network equilibria,”Mathematical Programming 31 (1985) 57–66.
 N.H. Josephy, “Newton's method for generalized equations,” Technical Report No. 1965, Mathematics Research Center, University of Wisconsin (Madison, WI, 1979).
 N.H. Josephy, “QuasiNewton methods for generalized equations,” Technical Report No. 1966, Mathematics Research Center, University of Wisconsin (Madison, WI, 1979).
 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).
 S. Karamardian, “The nonlinear complementarity problem with applications, parts I and II,”Journal of Optimization Theory and Applications 4 (1969) 87–98 and 16781.
 S. Karamardian, “Generalized complementarity problem,”Journal of Optimization Theory and Applications 8 (1971) 161–167.
 S. Karamardian, “The complementarity problem,”Mathematical Programming 2 (1972) 107–129.
 S. Karamardian, “Complementarity problems over cones with monotone and pseudomonotone maps,”Journal of Optimization Theory and Applications 18 (1976) 445–454.
 S. Karamardian, “An existence theorem for the complementarity problem,”Journal of Optimization Theory and Applications 18 (1976) 445–454.
 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).
 D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Application (Academic Press, New York, 1980).
 M. Kojima, “Computational methods for solving the nonlinear complementarity problem,”Keio Engineering Reports 27 (1974) 1–41.
 M. Kojima, “A unification of the existence theorems of the nonlinear complementarity problem,”Mathematical Programming 9 (1975) 257–277.
 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.
 M. Kojima, S. Mizuno, and T. Noma, “A new continuation method for complementarity problems with uniform Pfunctions,”Mathematical Programming 43 (1989) 107–114.
 M. Kojima, S. Mizuno, and T. Noma, “Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems,” Research Report No. B199, Department of Information Sciences, Tokyo Institute of Technology (Tokyo, Japan, 1988).
 M.M. Kostreva, “Block pivot methods for solving the complementarity problem,”Linear Algebra and Its Application 21 (1978) 207–215.
 M.M. Kostreva, “Elastohydrodynamic lubrication: a nonlinear complementarity problem,”International Journal for Numerical Methods in Fluids 4 (1984) 377–397.
 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).
 H.W. Kuhn, “Simplicial approximation of fixed points,”Proceedings of the National Academy of Sciences U.S.A. 61 (1968) 1238–1242.
 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.
 J. Kyparisis, “Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems,”Mathematical Programming 36 (1986) 105–113.
 J. Kyparisis, “Sensitivity analysis framework for variational inequalities,”Mathematical Programming 38 (1987) 203–213.
 J. Kyparisis, “Perturbed solutions of variational inequality problems over polyhedral sets,”Journal of Optimization Theory and Applications 57 (1988) 295–305.
 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).
 S. Lawphongpanich and D.W. Hearn, “Simplicial decomposition of asymmetric traffic assignment problem,”Transportation Research 18B (1984) 123–133.
 S. Lawphongpanich and D.W. Hearn, “Bender's decomposition for variational inequalities,”Mathematical Programming (Series B) 48 (1990) 231–247, this issue.
 S. Lawphongpanich and D.W. Hearn, “Restricted simplicial decomposition with application to the traffic assignment problem,”Ricera Operativa 38 (1986) 97–120.
 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.
 C.E. Lemke, “Bimatrix equilibrium points and mathematical programming,”Management Science 11 (1965) 681–689.
 C.E. Lemke and J.T. Howson, “Equilibrium points of bimatrix games,”SIAM Review 12 (1964) 45–78.
 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.
 J.L. Lions and G. Stampacchia, “Variational inequalities,”Communications on Pure and Applied Mathematics 20 (1967) 493–519.
 H.J. Lüthi, “On the solution of variational inequality by the ellipsoid method,”Mathematics of Operations Research 10 (1985) 515–522.
 T.L. Magnanti, “Models and algorithms for predicting urban traffic equilibria,” in: M. Florian, ed.,Transportation Planning Models (NorthHolland, Amsterdam, 1984) pp. 153–185.
 O. Mancino and G. Stampacchia, “Convex programming and variational inequalities,”Journal of Optimization Theory and Application 9 (1972) 3–23.
 O.L. Mangasarian, “Equivalence of the complementarity problem to a system of nonlinear equations,”SIAM Journal on Applied Mathematics 31 (1976) 89–92.
 O.L. Mangasarian, “Locally unique solutions of quadratic programs, linear and nonlinear complementarity problems,”Mathematical Programming 19 (1980) 200–212.
 O.L. Mangasarian and L. McLinden, “Simple bounds for solutions of monotone complementarily problems and convex programs,”Mathematical Programming 32 (1985) 32–40.
 A.S. Manne, “On the formulation and solution of economic equilibrium models,”Mathematical Programming Study 23 (1985) 1–22.
 A.S. Manne and P.V. Preckel, “A threeregion intertemporal model of energy, international trade and capital flows,”Mathematical Programming Study 23 (1985) 56–74.
 P. Marcotte, “Network optimization with continuous control parameters,”Transportation Science 17 (1983) 181–197.
 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.
 P. Marcotte, “A new algorithm for solving variational inequalities with application to the traffic assignment problem,”Mathematical Programming 33 (1985) 339–351.
 P. Marcotte, “Gapdecreasing algorithms for monotone variational inequalities,” paper presented at the ORSA/TIMS Meeting, Miami Beach, October 1986.
 P. Marcotte, “Network design with congestion effects: a case of bilevel programming,”Mathematical Programming 34 (1986) 142–162.
 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.
 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.
 L. Mathiesen, “Computation of economic equilibria by a sequence of linear complementarity problems,”Mathematical Programming Study 23 (1985) 144–162.
 L. Mathiesen, “Computational experience in solving equilibrium models by a sequence of linear complementarity problems,”Operations Research 33 (1985) 1225–1250.
 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.
 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).
 L. Mathiesen and E. Steigum, Jr., “Computation of unemployment equilibria in a twocountry multiperiod model with neutral money,” Working Paper, Center for Applied Research, Norwegian School of Economics and Business Administration (Bergen, Norway, 1985).
 L. McKenzie, “Why compute economic equilibria?,” in:Computing Equilibria: How and Why (NorthHolland, Amsterdam, 1976).
 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.
 L. McLinden, “An analogue of Moreau's proximation theorem, with application to the nonlinear complementarity problem,”Pacific Journal of Mathematics 88 (1980) 101–161.
 L. McLinden, “Stable monotone variational inequalities,”Mathematical Programming (Series B) 48 (1990) 303–338, this issue.
 N. Megiddo, “A monotone complementarity problem with feasible solutions but no complementary solutions,”Mathematical Programming 12 (1977) 131–132.
 N. Megiddo, “On the parametric nonlinear complementarity problem,”Mathematical Programming Study 7 (1978) 142–159.
 N. Megiddo and M. Kojima, “On the existence and uniqueness of solutions in nonlinear complementarity theory,”Mathematical Programming 12 (1977) 110–130.
 G.J. Minty, “Monotone (nonlinear) operators in Hilbert space,”Duke Mathematics Journal 29 (1962) 341–346.
 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).
 J.J. Moré, “Classes of functions and feasibility conditions in nonlinear complementarity problems,”Mathematical Programming 6 (1974) 327–338.
 J.J. Moré, “Coercivity conditions in nonlinear complementarity problems,”SIAM Review 17 (1974) 1–16.
 J.J. Moré and W.C. Rheinboldt, “On P and Sfunctions and related classes of ndimensional nonlinear mappings,”Linear Algebra and Its Applications 6 (1973) 45–68.
 J.J. Moreau, “Proximitè et dualitè dans un espace Hilberiten,”Bulletin of the Society of Mathematics of France 93 (1965) 273–299.
 J.D. Murchland, “Braess' paradox of traffic flow,”Transportation Research 4 (1970) 391–394.
 K.G. Murty,Linear Complementarity, Linear and Nonlinear Programming (Helderman, Berlin, 1988).
 A. Nagurney, “Comparative tests of multimodal traffic equilibrium methods,”Transportation Research 18B (1984) 469–485.
 A. Nagurney, “Computational comparisons of algorithms for general asymmetric traffic equilibrium problems with fixed and elastic demand,”Transportation Research 20B (1986) 78–84.
 A. Nagurney, “Computational comparisons of spatial price equilibrium methods,”Journal of Regional Science 27 (1987) 55–76.
 A. Nagurney, “Competitive equilibrium problems, variational inequalities and regional science,”Journal of Regional Science 27 (1987) 503–517.
 J.F. Nash, “Equilibrium points in nperson games,”Proceedings of the National Academy of Sciences 36 (1950) 48–49.
 S. Nguyen and C. Dupuis, “An efficient method for computing traffic equilibria in networks with asymmetric transportation costs,”Transportation Science 18 (1984) 185–202.
 J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
 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.
 J.S. Pang,LeastElement Complementarity Theory. Ph.D. dissertation, Department of Operations Research, Stanford University (Stanford, CA, 1976).
 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.
 J.S. Pang, “A column generation technique for the computation of stationary points,”Mathematics of Operations Research 6 (1981) 213–244.
 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.
 J.S. Pang, “Solution of the general multicommodity spatial equilibrium problem by variational and complementarity methods,”Journal of Regional Science 24 (1984) 403–414.
 J.S. Pang, “Variational inequality problems over product sets: applications and iterative methods,”Mathematical Programming 31 (1985) 206–219.
 J.S. Pang, “Inexact Newton methods for the nonlinear complementarity problem,”Mathematical Programming 36 (1986) 54–71.
 J.S. Pang, “A posteriori error bounds for linearly constrained variational inequality problems,”Mathematics of Operations Research 12 (1987) 474–484.
 J.S. Pang, “Two characterization theorems in complementarity theory,”Operations Research Letters 7 (1988) 27–31.
 J.S. Pang, “Newton's method for Bdifferentiable equations,” to appear in:Mathematics of Operations Research.
 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.
 J.S. Pang and D. Chan, “Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313.
 J.S. Pang and J.M. Yang, “Parallel Newton methods for the nonlinear complementarity problem,”Mathematical Programming (Series B) 42 (1988) 407–420.
 J.S. Pang and C.S. Yu, “Linearized simplicial decomposition methods for computing traffic equilibria on networks,”Networks 14 (1984) 427–438.
 P.V. Preckel, “Alternative algorithms for computing economic equilibria,”Mathematical Programming Study 23 (1985) 163–172.
 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.
 Y. Qiu and T.L. Magnanti, “Sensitivity analysis for variational inequalities defined on polyhedral sets,”Mathematics of Operations Research 14 (1989) 410–432.
 Y. Qiu and T.L. Magnanti, “Sensitivity analysis for variational inequalities,” Working Paper OR 16387, Operations Research Center, M.I.T. (Cambridge, MA, 1987).
 A. Reinoza,A Degree For Generalized Equations. Ph.D. dissertation, Department of Industrial Engineering, University of Wisconsin (Madison, WI, 1979).
 A. Reinoza, “The strong positivity conditions,”Mathematics of Operations Research 10 (1985) 54–62.
 W.C. Rheinboldt,Numerical Analysis of Parameterized Nonlinear Equations (Wiley, New York, 1986).
 S.M. Robinson, “Generalized equations and their solutions, part I: basic theory,”Mathematical Programming Study 10 (1979) 128–141.
 S.M. Robinson, “Strongly regular generalized equations,”Mathematics of Operations Research 5 (1980) 43–62.
 S.M. Robinson, “Generalized equations and their solutions, part II: applications to nonlinear programming,”Mathematical Programming Study 19 (1982) 200–221.
 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.
 S.M. Robinson, “Implicit Bdifferentiability in generalized equations,” Technical Summary Report No. 2854, Mathematics Research Center, University of Wisconsin (Madison, WI, 1985).
 S.M. Robinson, “Local structure of feasible sets in nonlinear programming, part III: stability and sensitivity,”Mathematical Programming Study 30 (1987) 45–66.
 S.M. Robinson, “An implicitfunction theorem for a class of nonsmooth functions,” to appear in:Mathematics of Operations Research.
 R.T. Rockafellar, “Characterization of the subdifferentials of convex functions,”Pacific Journal of Mathematics 17 (1966) 497–510.
 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.
 R.T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,”Pacific Journal of Mathematics 33 (1970) 209–216.
 R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
 R.T. Rockafellar, “Augmented Lagrangian and application of the proximal point algorithm in convex programming,”Mathematics of Operations Research 1 (1976) 97–116.
 R.T. Rockafellar, “Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization 14 (1976) 877–898.
 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.
 T.F. Rutherford,Applied General Equilibrium Modeling. Ph.D. dissertation Department of Operations Research, Stanford University (Stanford, CA, 1986).
 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).
 R. Saigal, “Extension of the generalized complementarity problem,”Mathematics of Operations Research 1 (1976) 260–266.
 P.A. Samuelson, “Spatial price equilibrium and linear programming,”American Economic Review 42 (1952) 283–303.
 H.E. Scarf, “The approximation of fixed points of a continuous mapping,”SIAM Journal on Applied Mathematics 15 (1967) 1328–1342.
 H.E. Scarf and T. Hansen,Computation of Economic Equilibria (Yale University Press, New Haven, CT, 1973).
 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).
 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.
 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.
 S. Smale, “A convergent process of price adjustment and global Newton methods,”Journal of Mathematical Economics 3 (1976) 107–120.
 M.J. Smith, “The existence, uniqueness and stability of traffic equilibria,”Transportation Research 13B (1979) 295–304.
 M.J. Smith, “The existence and calculation of traffic equilibria,”Transportation Research 17B (1983) 291–303.
 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.
 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.
 T.E. Smith, “A solution condition for complementarity problems: with an apilication to spatial price equilibrium,”Applied Mathematics and Computation 15 (1984) 61–69.
 J.E. Spingarn, “Partial inverse of a monotone operator,”Applied Mathematics and Optimization 10 (1983) 247–265.
 J.E. Spingarn, “Applications of the method of partial inverses to convex programming: decomposition,”Mathematical Programming 32 (1985) 199–223.
 J.E. Spingarn, “On computation of spatial economic equilibria,” Discussion Paper 8731, Center for Operations Research and Econometrics, Université Catholique de Louvain (LouvainlaNeuve, Belgium, 1987).
 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.
 R. Steinberg and R.E. Stone, “The prevalence of paradoxes in transportation equilibrium problems,” Working paper, AT&T Bell Laboratories (Holmdel, NJ, 1987).
 R. Steinberg and W.I. Zangwill, “The prevalence of Braess' paradox,”Transportation Science 17 (1983) 301–319.
 J.C. Stone, “Sequential optimization and complementarity techniques for computing economic equilibria,”Mathematical Programming Study 23 (1985) 173–191.
 P.K. Subramanian, “GaussNewton methods for the nonlinear complementarity problem,” Technical Summary Report No. 2845, Mathematics Research Center, University of Wisconsin (Madison, WI, 1985).
 P.K. Subramanian, “Fixedpoint methods for the complementarity problem,” Technical Summary Report No. 2857, Mathematics Research Center, University of Wisconsin (Madison, WI, 1985).
 P.K. Subramanian, “A note on least two norm solutions of monotone complementarity problems,”Applied Mathematics Letters 1 (1988) 395–397.
 A. Tamir, “Minimality and complementarity properties associated with Zfunctions and Mfunctions,”Mathematical Programming 7 (1974) 17–31.
 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.
 R.L. Tobin, “Sensitivity analysis for variational inequalities,”Journal of Optimization Theory and Applications 48 (1986) 191–204.
 M.J. Todd,The Computation of Fixed Points and Applications (Springer, Berlin, 1976).
 M.J. Todd, “A note on computing equilibria in economics with activity models of production“,Journal of Mathematical Economics 6 (1979) 135–144.
 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.
 J.A. Ventura and D.W. Hearn, “Restricted simplicial decomposition for convex constrained problems,” Research Report No. 8615, Department of Industrial and Systems Engineering, University of Florida (Gainesville, FL, 1986).
 J.G. Wardrop, “Some theoretical aspects of road traffic research,”Proceedings of the Institute of Civil Engineers, Part II (1952) 325–378.
 L.T. Watson, “Solving the nonlinear complementarity problem by a homotopy method,”SIAM Journal on Control and Optimization 17 (1979) 36–46.
 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.
 Y. Yamamoto, “A path following algorithm for stationary point problems,”Journal of the Operations Research Society of Japan 30 (1987) 181–198.
 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.
 Title
 Finitedimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications
 Journal

Mathematical Programming
Volume 48, Issue 13 , pp 161220
 Cover Date
 19900301
 DOI
 10.1007/BF01582255
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Variational inequality
 complementarity
 fixed points
 Walrasian equilibrium
 traffic assignment
 network equilibrium
 spatial price equilibrium
 Nash equilibrium
 Industry Sectors
 Authors

 Patrick T. Harker ^{(1)}
 JongShi Pang ^{(2)}
 Author Affiliations

 1. Decision Sciences Department, The Wharton School, University of Pennsylvania, 191046366, Philadelphia, PA, USA
 2. Department of Mathematical Sciences, The Whiting School of Engineering, The Johns Hopkins University, 21218, Baltimore, MD, USA