Handbook of Global Optimization pp 271-338 | Cite as
Complementarity Problems
Chapter
Abstract
This chapter presents a comprehensive treatment of the nonlinear complementarity problem and several related mathematical programs in finite dimensions. Topics discussed include existence theory, solution methods, sensitivity and stability analysis, and applications to equilibrium modeling and engineering problems. Some future research directions are suggested and an extensive list of references is given.
Key words:
Complementarity problems Variational inequalities Nonlinear programming Equilibrium modeling Global optimization Nonsmooth equations.Preview
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References
- 1.A1-Khayyal, F.A.: 1986, “Linear, quadratic, and bilinear programming approaches to the linear complementarity problem”, European Journal of Operational Research Vol. no. 86, pp. 216–227.Google Scholar
- 2.A1-Khayyal, F.A.: 1990, “On solving linear complementarity problems as bilinear programs”, Arabian Journal for Science and Engineering Vol. no. 15, pp. 639–646.Google Scholar
- 3.Anderson, E.J. and Wu, S.Y.: 1991, “The continuous complementarity problem”, Optimization Vol. no. 22, pp. 419–426.MathSciNetMATHGoogle Scholar
- 4.Aashtiani, H.Z. and Magnanti, T.L.: 1982, “Equilibria on a congested transportation network”, SIAM Journal on Algebraic and Discrete Methods Vol. no. 2, pp. 213–226.MathSciNetGoogle Scholar
- 5.Aubin, J.-P. and Frankowska, H.: 1990, Set-Valued Analysis, Birkäuser, Boston.MATHGoogle Scholar
- 6.Auclunuty, G.: 1989, “Variational principles for variational inequalities”, Numerical Functional Analysis and Optimization Vol. no. 10, pp. 863–874.Google Scholar
- 7.Balocchi, C. and Capelo, A.: 1984, Variational and Quasivariational Inequalities: Applications to free boundary problems, John Wiley and Sons, Chichester.Google Scholar
- 8.Bank, H., Guddat, J., Klatte, D., Kummer, B., and Tammer, K.: 1982, Non-Linear Parametric Optimization, Birkhäuser Berlag, Basel.Google Scholar
- 9.Björkman, G.: 1991, “The solution of large displacement frictionless contact problems using a sequence of linear complementarity problems”, International Journal for Numerical Methods in Engineering Vol. no. 31, pp. 1553–1556.MATHGoogle Scholar
- 10.Bonpans, J.F.: 1990, “Rates of convergence of Newton type methods for variational inequalities and nonlinear programming”, manuscript, INRIA.Google Scholar
- 11.Cao, M. and Ferris, M.C.: 1992, “An interior point algorithm for monotone affine variational inequalities”, Technical report, Computer Sciences Department, University of Wisconsin, Madison.Google Scholar
- 12.Cao, M. and Ferris, M.C.: 1992, “A pivotal method for affine variational inequalities”, Technical report #1114, Computer Sciences Department, University of Wisconsin, Madison.Google Scholar
- 13.Chan, D. and Pang, J.S.: 1982, “The generalized quasi-variational inequality problem”, Mathematics of Operations Research Vol. no. 7, pp. 211–222.MathSciNetMATHGoogle Scholar
- 14.Cheng, Y.C.: 1984, “On the gradient-projection method for solving the nonsymmetric linear complementarity problem”, Journal of Optimization Theory and Applications Vol. no. 43, pp. 527–541.MATHGoogle Scholar
- 15.Chipot, M.: 1984, Variational Inequalities and Flow in Porous Media, Applied Mathematical Sciences, Vol. no. 52, Springer Verlag, New York.Google Scholar
- 16.Corni, C. and Maier, G.: 1990, “Extremum theorem and convergence criterion for an iterative solution to the finite-step problem in elastoplasticity with mixed nonlinear hardening”, European Journal of Mechanics A/Solids Vol. no. 9, pp. 563–585.Google Scholar
- 17.Cottle, R.W.: 1964, “Nonlinear programs with positively bounded Jacobians”, Ph.D. thesis, Department of Mathematics, University of California, Berkeley.Google Scholar
- 18.Cottle, R.W.: 1966, “Nonlinear programs with positively bounded Jacobians”, SIA M Journal on Applied Mathematics Vol. no. 14, pp. 1472013158.MathSciNetMATHGoogle Scholar
- 19.Cottle, R.W., Kyparisis, J., and Pang, J.S. (eds.): 1990, Variational Inequality Problems, North-Holland, The Netherlands [also published as Mathematical Programming, Series B, Vol. no. 48 (2)].Google Scholar
- 20.Cottle, R.W., Kyparisis, J., and Pang, J.S. (eds.): 1990, Complementarity Problems, North-Holland, The Netherlands [also published as Mathematical Programming, Series B, Vol. no. 48 (3)].Google Scholar
- 21.Cottle, R.W., Pang, J.S., and Stone, R.E.: 1992, The Linear Complementarity Problem, Academic Press, Boston.MATHGoogle Scholar
- 22.Cottle, R.W., Pang, J.S., and Venkateswaran, V.: 1989, “Sufficient matrices and the linear complementarity problem”, Linear Algebra and its Applications Vol. no. 114/115, pp. 231–249.MathSciNetGoogle Scholar
- 23.Dafermos, S.C.: 1988: “Sensitivity analysis in variational inequalities”, Mathmematics of Operations Research Vol. no. 13, pp. 421–434.MathSciNetMATHGoogle Scholar
- 24.Dafermos, S.C. and McKelvey, S.C.: 1992, “Partitionable variational inequalities with applications to network and economic equilibria”, Journal of Optimization Theory and Applications Vol. no. 73, pp. 243–268.MathSciNetMATHGoogle Scholar
- 25.Dantzig, G.B. and Manne, A.S.: 1974, “A complementarity algorithm for an optimal capital path with invariant proportions”, Journal of Economic Theory Vol. no. 9, pp. 312–323.Google Scholar
- 26.De Donato, O. and Maier, G.: 1972, “Mathematical programming methods for the inelastic analysis of reinforced concrete frames allowing for limited rotation capacity”, International Journal for Numerical Methods in Engineering Vol. no. 4, pp. 307–329.Google Scholar
- 27.De Moor, B.: 1988, “Mathematical concepts and techniques for modelling of static and dynamic systems”, Ph.D. dissertation, Departement Elektrotechniek, Katholieke Universiteit Leuven, Leuven, Belgium.Google Scholar
- 28.Dennis, J.E., Jr. and Schnabel, R.B.: 1983, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, New Jersey.MATHGoogle Scholar
- 29.Dontchev, A.L. and Hager, W.W.: 1993, “Lipschitzian stability in nonlinear control and optimization”, SIAM Journal on Control and Optimization Vol. no. 31, pp. 569–603.MathSciNetMATHGoogle Scholar
- 30.Dontchev, A.L. and Hager, W.W.: 1992, “Implicit functions, Lipschitz maps, and stability in optimization”, manuscript, Department of Mathematics, University of Florida, Gainesville.Google Scholar
- 31.Eaves, B.C.: 1971, “On the basic theorem of complementarity”, Mathematical Programming Vol. no. 1, pp. 68–75.MathSciNetMATHGoogle Scholar
- 32.Eaves, B.C.: 1972, “Homotopies for computation of fixed points”, Mathematical Programming Vol. no. 3, pp. 1–22.MathSciNetMATHGoogle Scholar
- 33.Eaves, B.C.: 1976, “A short course in solving equations with PL homotopies”, in R.W. Cottle and C.E. Lemke, eds., Nonlinear Programming: SIAM-AMS Proceedings 9, American Mathematical Society, Providence, pp. 73–143.Google Scholar
- 34.Eaves, B.C.: 1983, “Where solving for stationary points by LCPs is mixing Newton iterates”, in B.C. Eaves, F.J. Gould, H.O. Peitgen, and M.J. Todd, eds., Homotopy Methods and Global Convergence, Plenum Press, New York, pp. 63–78.Google Scholar
- 35.Ferris, M.C. and Mangasarian, O.L.: 1991, “Error bounds and strong upper semicontinuity for monotone affine variational inequalities”, Technical report #1056, Computer Sciences Department, University of Wisconsin, Madison; Annals of Operations Research, forthcoming.Google Scholar
- 36.Fiacco, A.V.: 1983, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York.MATHGoogle Scholar
- 37.Fiacco, A.V. and McCormick, G.P.: 1968, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, New York.MATHGoogle Scholar
- 38.Florian, M.: 1989, Mathematical programming applications in national, regional and urban planning, in M. Iri and K. Tanabe, eds., Mathematical Programming: Recent Developments and Applications, Kluwer Academic Publishers, Tokyo, pp. 57–82.Google Scholar
- 39.Friedman, A.: 1982, Variational Principles and Free-Boundary Problems, John Wiley and Sons, New York.MATHGoogle Scholar
- 40.Friesz, T.L. and Harker, P.T.: 1985, “Freight network equilibrium: a review of the state of the art”, in A. Daughety, ed., Analytical Studies in Transportation Economics, Cambridge University Press, Cambridge, pp. 161–206.Google Scholar
- 41.Fujisawa, T. and Kuh, E.S.: 1972, “Piecewise-linear theory of resistive networks”, SIAM Journal on Applied Mathematics Vol. no. 22, pp. 307–328.MathSciNetMATHGoogle Scholar
- 42.Fukushima, M.: 1992, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems”, Mathematical Programming Vol. no. 53, pp. 45–62.MathSciNetGoogle Scholar
- 43.Gabriel, S.A. and Pang, J.S.: 1992, “An inexact NE/SQP method for solving the nonlinear complementarity problem”, Computational Optimization and Applications Vol. no. 1, pp. 67–91.MathSciNetMATHGoogle Scholar
- 44.Gabriel, S.A. and Pang, J.S.: 1993, “A trust region method for constrained nonsmooth equations”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
- 45.Gauvin, J.: 1977, “A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming”, Mathematical Programming Vol. no. 12, pp. 136–138.MathSciNetMATHGoogle Scholar
- 46.Garica, C.B. and Zangwill, W.I.: 1981, Pathways to Solutions, Fixed Points, and Equilibria, Prentice-Hall, Englewood Cliffs.Google Scholar
- 47.Gowda, M.S.: 1989, “ Pseucjomonotone and copositive-star matrices”, Linear Algebra and its Applications Vol. no. 113, pp. 107–110.MathSciNetMATHGoogle Scholar
- 48.Gowda, M.S.: 1989, “Complementarity problems over locally compact cones”, SIAM Journal on Control and Optimization Vol. no. 27, pp. 836–841.MathSciNetMATHGoogle Scholar
- 49.Gowda, M.S. and Pang, J.S.: 1992, “On solution stability of the linear complementarity problem”, Mathematics of Operations Research Vol. no. 17, pp. 77–83.MathSciNetMATHGoogle Scholar
- 50.Gowda, M.S. and Pang, J.S.: 1993, “Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory”, Mathematics of Operations Research, forthcoming.Google Scholar
- 51.Gowda, M.S. and Pang, J.S.: 1993, “On the boundedness and stability of solutions to the affine variational inequality problem”, SIAM Journal on Control and Optimization, forthcoming.Google Scholar
- 52.Gowda, M.S. and Seidman, T.I.: 1990, “Generalized linear complementarity problem”, Mathematical Programming Vol. no. 46, pp. 329–340.MathSciNetMATHGoogle Scholar
- 53.Gowda, M.S., and Sznajder, R.: 1993, “The generalized order linear complementarity problem”, SIAM Journal on Matrix Analysis and Applications Vol. no. 14, pp.Google Scholar
- 54.Guddat, J., Vasquez, F.G., with Jongen, H.Th.: 1990, Parametric Optimization: Singularities, Pathfollowing and Jumps, John Wiley & Sons, Chichester.Google Scholar
- 55.Güler, 0.: 1993, “Existence of interior points and interior paths in nonlinear complementarity problems”, Mathematics of Operations Research Vol. no. 18, pp. 128–147.MathSciNetMATHGoogle Scholar
- 56.Gupta, S. and Pardalos, P.M.: 1988, “A note on a quadratic formulation for linear complementarity problems”, Journal of Optimization Theory and Applications Vol. no. 57, pp. 197–201.MathSciNetMATHGoogle Scholar
- 57.Ha, C.D.: 1987, “Application of degree theory in stability of the complementarity problem”, Mathematics of Operations Research Vol. no. 31, 327–338.Google Scholar
- 58.Hansen, T. and Koopmans, T.C.: 1972, “On the definition and computation of a capital stock invariant under optimization”, Journal of Economic Theory Vol. no. 5, pp. 487–523.MathSciNetGoogle Scholar
- 59.Haraux, A.: 1977,“How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities”, Journal of the Mathematical Society of Japan Vol. no. 29, pp. 615–631.MathSciNetMATHGoogle Scholar
- 60.Harker, P.T.: 1987, Predicting Intercity Freight Flows, VNU Science Press, Utrecht.Google Scholar
- 61.Harker, P.T. and Pang, J.S.: 1990, “Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications”, Mathematical Programming, Series B Vol. no. 48, pp. 161–220.MathSciNetMATHGoogle Scholar
- 62.Harker, P.T. and Xiao, B: 1990, “Newton’s method for the nonlinear complementarity problem: a B-differentiable equation approach”, Mathematical Programming, Series B Vol. no. 48, pp. 339–357.MathSciNetMATHGoogle Scholar
- 63.Harrison, M.J. and Reiman, M.I.: 1981, “Reflected Brownian motion on an orthant”, The Journal of Probability Vol. no. 9, pp. 302–308.MathSciNetMATHGoogle Scholar
- 64.Hartman, P. and Stampacchia, G.: 1966, “On some nonlinear elliptic differential functional equations”, Acta Mathematica Vol. no. 15, pp. 153–188.Google Scholar
- 65.Hearn, D.W.: 1982, “The gap function of a convex program”, Operations Research Letters Vol. no. 1, pp. 67–71.MathSciNetMATHGoogle Scholar
- 66.Hlavâeek, I., Haslinger, J., News, J., and Lovisek, J.: 1988, Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences, Vol. no. 66, Springer-Verlag, New York.Google Scholar
- 67.Hoffman, A.J.: 1952, “On approximate solutions of systems of linear inequalities”, Journal of Research of the National Bureau of Standards Vol. no. 49, pp. 263–265.Google Scholar
- 68.Horst, R. and Tuy, H.: 1990, Global Optimization, Springer Verlag, New york.MATHGoogle Scholar
- 69.Ip, C.M. and Kyparisis, J., “Local convergence of quasi-Newton methods for B-differentiable equations”, Mathematical Programming Vol. no 56, pp. 71–90.Google Scholar
- 70.Isac, G.: 1992, Complementarily Problems, Lecture Notes in Mathematics, Springer Verlag, New York.Google Scholar
- 71.Isac, G. and Kostreva, M.M.: 1991, “The generalized order complementarity problem”, Journal of Optimization Theory and Applications Vol. no. 71, pp. 517–534.MathSciNetMATHGoogle Scholar
- 72.Istratescu, V.I.: 1981, Fixed Point Theory, D. Reidel Publishing Company, Boston.MATHGoogle Scholar
- 73.Jones, P.C.: 1977, “Calculation of an optimal invariant stock”, Ph.D. thesis, Department of Industrial Engineering, University of California, Berkeley.Google Scholar
- 74.Josephy, N.H.: 1979, “Newton’s method for generalized equations”, Technical report #1965, Mathematics Research Center, University of Wisconsin, Madison.Google Scholar
- 75.Josephy, N.H.: 1979, “Quasi-Newton methods for generalized equations”, Technical Report No. 1965, Mathematics Research Center, University of Wisconsin, Madison.Google Scholar
- 76.Kanzow, C.: 1993, “Some equation-based methods for the nonlinear complementarity problem”, Preprint 63, Institut far Angewandte Mathematik, Universität Hamburg, Hamburg, Germany.Google Scholar
- 77.Kanzow, C.: 1993, “Nonlinear complementarity as unconstrained optimization”, Preprint 67, Institut für Angewandte Mathematik, Universität Hamburg, Hamburg, Germany.Google Scholar
- 78.Karamardian, S.: 1969, “The nonlinear complementarity problem with applications, part 1”, Journal of Optimization Theory and Applications Vol. no. 4, pp. 87–98.MathSciNetMATHGoogle Scholar
- 79.Karamardian, S.: 1971, “Generalized complementarity problems”, Journal of Optimization Theory and Applications Vol. no. 19, pp. 161–168.MathSciNetGoogle Scholar
- 80.Karamardian, S.: 1972, “The complementarity problem”, Mathematical Programming Vol. no. 2, pp. 107–129.MathSciNetMATHGoogle Scholar
- 81.Kinderlehrer, D. and Stampacchia, G.: 1980, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York.MATHGoogle Scholar
- 82.King, A.J. and Rockafellâr, R.T.: 1992, “Sensitivity analysis for nonsmooth generalized equations”, Mathematical Programming Vol. no. 55, pp. 193–212.MathSciNetMATHGoogle Scholar
- 83.Klarbring, A. and Björkman, G.: 1992, “Solution of large displacement contact problems with friction using Newton’s method for generalized equations”, International Journal for Numerical Methods in Engineering Vol. no. 34, pp. 249–269.MATHGoogle Scholar
- 84.Kojima, M.: 1980, “Strongly stable stationary solutions in nonlinear programs”, in S.M. Robinson, ed., Analysis and Computation of Fixed Points, Academic Press, New York, pp. 93–138.Google Scholar
- 85.Kojima, M. and Hiabayashi, R.: 1984, “Continuous deformation of nonlinear programs”, Mathematical Programming Study Vol. no. 21, pp. 150–198.MATHGoogle Scholar
- 86.Kojima, M., Megiddo, N., and Noma, T.: 1991, “Homotopy continuation methods for nonlinear complementarity problems”, Mathematics of Operations Research Vol. no. 16, pp. 754–774.MathSciNetMATHGoogle Scholar
- 87.Kojima, M., Megiddo, N., Noma, T., and Yoshise, A.: 1991, A Unified Approach to Interior Point Algorithms for Linear Complementarily Problems, Lecture Notes in Computer Science 538, Springer Verlag, Berlin.Google Scholar
- 88.Kojima, M., Mizuno, S., and Noma, T.: 1989, “A new continuat:on method for complementarity problems with uniform P-functions”, Mathematical Programming Vol. no. 43, pp. 107–113.MathSciNetMATHGoogle Scholar
- 89.Kojima, M., Mizuno, S., and Noma, T.: 1990, “Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems”, Mathematics of Operations Research Vol. no. 15, pp. 662–675.MathSciNetMATHGoogle Scholar
- 90.Kojima, M., Noma, T., and Yoshise, A.: 1992, “Global convergence and detecting infeasibility in interior-point algorithms”, Research report B-257, Department of Information Sciences, Tokyo Institute of Technology, Tokyo.Google Scholar
- 91.Kostreva, M.M., 1978, “Block pivot methods for solving the complementarity problem”, Linear Algebra and its Applications Vol. no. 21, pp. 207–215.MathSciNetMATHGoogle Scholar
- 92.Kostreva, M.M.: 1984, “Elasto-hydrodynamic lubrication: a nonlinear complementarity problem”, International Journal for Numerical Methods in Fluids Vol. no. 4, pp. 377–397.MathSciNetMATHGoogle Scholar
- 93.Kyparisis, J.: 1985, “On uniqueness of Kuhn-Tucker multipliers in nonlinear programming”, Mathematical Programming Vol. no. 32, pp. 242–246.MathSciNetMATHGoogle Scholar
- 94.Kyparisis, J.: 1986, “Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems”, Mathematical Programming Vol. no. 36, pp. 105–113.MathSciNetMATHGoogle Scholar
- 95.Kyparisis, J.: 1990, “Sensitivity analysis for variational inequalities and nonlinear complementarity problems”, Annals of Operations Research Vol. no. 27, 143–174.MathSciNetMATHGoogle Scholar
- 96.Kyparisis, J.: 1990, “Solution differentiability for variational inequalities”, Mathematical Programming, Series B Vol. no. 48, pp. 285–302.MathSciNetMATHGoogle Scholar
- 97.Kyparisis, J.: 1992, “Parametric variational inequalities with multivalued solution sets”, Mathematics of Operations Vol. no. 17, pp. 341–364.MathSciNetMATHGoogle Scholar
- 98.Kyparisis, J. and Ip, C.M.: 1992, “Solution behavior for parametric implicit complementarity problems”, Mathematical Programming Vol. no. 56, pp. 71–90.MathSciNetMATHGoogle Scholar
- 99.Larsson, T. and Patriksson, M.: 1992, “A class of gap functions for variational inequalities”, manuscript, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden.Google Scholar
- 100.Lemke, C.E. and Howson, J.T., Jr.: 1964, “Equilibrium points of bimatrix games”, SIAM Journal on Applied Mathematics Vol. no. 12, pp. 413–423.MathSciNetMATHGoogle Scholar
- 101.Lin, Y.Y. and Pang, J.S.: 1987, “Iterative methods for large convex quadratic programs: A survey”, SIAM Journal on Control and Optimization Vol. no. 25, pp. 383–411.MathSciNetMATHGoogle Scholar
- 102.Lloyd, N.G.: 1978, Degree Theory, Cambridge University Press, Cambridge.Google Scholar
- 103.Lojasiewicz, M.S.: 1958, “Division d’une distribution par une fonction analytique de variables réelles”, Comptes de Rendus de Séance, Paris Vol. no. 146, pp. 683–686.MathSciNetGoogle Scholar
- 104.Lojasiewicz, M.S.: 1959, “Sur la problème de la division”, Studio Mathematica Vol. no. 18, pp. 87–136.MathSciNetMATHGoogle Scholar
- 105.Luo, X.D. and Luo, Z.Q.: 1992, “Extension of Hoffman’s error bound to polynomial systems”, Technical report, Communications Research Laboratory, McMaster University, Hamilton, Canada.Google Scholar
- 106.Luo, Z.Q.: 1992, “Convergence analysis of primal-dual interior point algorithms for solving convex quadratic programs”, Technical report, Communications Research Laboratory, McMaster University, Hamilton, Canada.Google Scholar
- 107.Luo, Z.Q., Mangasarian, O.L., Ren, J., and Solodov, M.V.: 1994, “New error bounds for the linear complementarity problem”, Mathematics of Operations Research Vol. no. 19, pp.Google Scholar
- 108.Luo, Z.Q. and Pang, J.S.: 1993, “Error bounds for analytic systems and their applications”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
- 109.Luo, Z.Q. and Tseng, P.: 1992: “Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem”,SIAM Journal on Optimization Vol. no. 2, pp. 43–54.MathSciNetMATHGoogle Scholar
- 110.Luo, Z.Q. and Tseng, P.: 1992: “On global error bound for a class of monotone affine variational inequality problems”, Operations Research Letters Vol. no. 11, pp. 159–165.MathSciNetMATHGoogle Scholar
- 111.Luo, Z.Q. and Tseng, P.: 1993, “Error bound and reduced gradient-projection algorithms for convex minimization over a polyhedral set”, SIAM Journal on Optimization Vol. no. 3, pp. 43–60.MathSciNetMATHGoogle Scholar
- 112.Lustig, I.J.: 1991, “Feasibility issues in a primal-dual interior-point method for linear programrning”, Mathematical Programming Vol. no. 49, pp. 145–162.MathSciNetGoogle Scholar
- 113.Magnanti, T.L.: 1984, “Models and algorithms for predicting urban traffic equilibrium”, in M. Florian, ed., Transportation Planning Models, North-Holland, Amsterdam, pp. 153–186.Google Scholar
- 114.Maier, G.: 1970, “A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes”, Meccanica Vol. no. 5, pp. 54–66.MATHGoogle Scholar
- 115.Maier, G. and Novati, G.: 1989, “A shakedown and bounding theory allowing for nonlinear hardening and second order geometric effects with reference to discrete structural models”, in J.A. König, ed., A Volumne in honor of A. Sawczuk.Google Scholar
- 116.Mandelbaum, A.: 1989, “The dynamic complementarity problem”, manuscript, Graduate School of Business, Stanford University, Stanford.Google Scholar
- 117.Mangasarian, O.L.: 1969, Nonlinear Programming, McGraw-Hill, New York.Google Scholar
- 118.Mangasarian, O.L.: 1976, “Equivalence of the complementarity problem to a system of nonlinear equations”, SIAM Journal on Applied Mathematics Vol. no. 31, pp. 89–992.MathSciNetMATHGoogle Scholar
- 119.Mangasarian, O.L.: 1980, “Locally unique solutions of quadratic programs, linear and nonlinear complementarity problems”, Mathematical Programming Vol. no. 19, pp. 200–212.MathSciNetMATHGoogle Scholar
- 120.Mangasarian, O.L.: 1992, “Global error bounds for monotone affine variational inequality problems”, Linear Algebra and its Applications Vol. no. 174, pp. 153–164.MathSciNetMATHGoogle Scholar
- 121.Mangasarian, O.L. and Fromovitz, S.: 1967, “The Fritz John optimality necessary conditions in the presence of equality and inequality constraints”, Journal of Mathematical Analysis and Applications Vol. no. 17, pp. 37–47.MathSciNetMATHGoogle Scholar
- 122.Mangasarian, O.L. and Shiau, T.H.: 1986, “Error bounds for monotone linear complementarity problems”, Mathematical Programming Vol. no. 36 pp. 81–89.MathSciNetMATHGoogle Scholar
- 123.Mangasarian, O.L. and Shiau, T.H.: 1986, “Error bounds for monotone linear complementarity problems”, Mathematical Programming Vol. no. 36 pp. 81–89.Google Scholar
- 124.Manne, A.S. (ed.): 1985, Economic Equilibrium: Model Formulation and Solution, North-Holland, Amsterdam [same as Mathematical Programming Study Vol. no. 231.Google Scholar
- 125.Marcotte, P.: 1985, “A new algorithm for solving variational inequalities, with application to the traffic assignment problem”, Mathematical Programming Vol. no. 33, pp. 339–351.MathSciNetGoogle Scholar
- 126.Marcotte, P. and Dussault, J.-P.: 1985, “A modified Newton method for solving variational inequalities”, Proceedings of the 24th IEEE Conference on Decision and Control, pp. 1433–1436.Google Scholar
- 127.Marcotte, P. and Dussault, J.-P.: 1987, “A note on a globally convergent Newton method for solving monotone variational inequalities”, Operations Research Letters Vol. no. 6, pp. 35–42.MathSciNetMATHGoogle Scholar
- 128.McCormick, G.P.: 1983, Nonlinear Programming: Theory, Algorithms, and Applications, John Wiley & Sons, New York.Google Scholar
- 129.McLinden, L.: 1980, “The complementarity problem for maximal monotone multifunction”, in R.W. Cottle, F. Giannessi, and J.L. Lions, eds., Variational Inequalities and Complementarity Problems, John Wiley, pp. 251–270.Google Scholar
- 130.Megiddo, N. and Kojima, M.: 1977, “On the existence and uniqueness of solutions in nonlinear complementarity problems”, Mathematical Programming Vol. no. 12, pp. 110–130.MathSciNetMATHGoogle Scholar
- 131.Minty, G.J.: 1962, “Monotone (non-linear) operators in Hilbert space”, Duke Mathematics Journal Vol. no. 29, pp. 341–346.MathSciNetMATHGoogle Scholar
- 132.Mittelman, H.D.: 1990, “Nonlinear parametrized equations: new results for variational problems and inequalities”, in E.L. Allgower and K. Georg, eds., Computational Solution of Nonlinear Systems of Equations, Lectures in Applied Mathematics, Vol. no. 26, American Mathematical Society, Providence, pp. 451–466.Google Scholar
- 133.Monteiro, R.D.C., Pang, J.S., and Wang, T.: 1992, “A positive algorithm for the nonlinear complementarity problem”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
- 135.Mordukhovich, B.: 1992, “Sensitivity analysis in nonsmooth optimization”, in D.A. Field & V. Komkov, eds., Theoretical Aspects of Industrial Design, SIAM Proceedings in Applied Mathematics, Vol. no. 58, pp. 32–46.Google Scholar
- 137.Mordukhovich, B.: 1992, “Lipschitzian stability of constraint systems and generalized equations”, manuscript, Department of Mathematics, Wayne State University, Detroit.Google Scholar
- 138.Moré, J. and Rheinboldt, W.C.: 1973, “On P- and S-functions and related class of n-dimensional nonlinear mappings”, Linear Algebra and its Applications Vol. no. 6, pp. 45–68.MATHGoogle Scholar
- 139.Murty, K.G.: 1988, Linear Complementarily, Linear and Nonlinear Programming, Helder-mann Verlag, Berlin.Google Scholar
- 140.Nagurney, A.: 1987, “Competitive equilibrium problems, variational inequalities and regional science”, Journal of Regional Science Vol. no. 27, pp. 55–76.Google Scholar
- 141.Nash, J.F.: 1950, “Equilibrium points in n-person games”, Proceedings of the National Academy of Sciences Vol. no. 36, pp. 48–49.MathSciNetMATHGoogle Scholar
- 142.Oh, K.P.: 1984, “The numerical solution of a dynamically loaded elastohydrodynamic contact as a nonlinear complementarity problem”, Transections of the ASME Vol. no. 106, pp. 88–94.Google Scholar
- 143.Oh, K.P.: 1986, “The formulation of the mixed lubrication problem as a generalized nonlinear complementarity problem”, Transactions of the ASME Vol. no. 108, pp. 598–604.Google Scholar
- 144.Ortega, J.M. and Rheinboldt, W.C.: 1970, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York.Google Scholar
- 145.Panagiotopoulos, P.D.: 1985, Inequality Problems in Mechanics and Applications, Birkhäuser, Boston.Google Scholar
- 146.Pang, J.S.: 1981, “The implicit complementarity problems”, in O.L. Mangasarian, R.R. Meyer, and S.M. Robinson, eds., Nonlinear Programming 4, Academic Press, New York, pp. 487–518.Google Scholar
- 147.Pang, J.S.: 1985, “Asymmetric variational inequality problems over product sets: applications and iterative methods”, Mathematical Programming Vol. no. 31, pp. 206–219.Google Scholar
- 148.Pang, J.S.: 1986, “Inexact Newton methods for the nonlinear complementarity problem”, Mathematical Programming Vol. no. 36, pp. 54–71.MATHGoogle Scholar
- 149.Pang, J.S.: 1987, “A posteriori error bounds for the linearly-constrained variational inequality problem”, Mathematics of Operations Research Vol. no. 12, pp. 474–484.Google Scholar
- 150.Pang, J.S.: 1990, “Newton’s method for B-differentiable equations”, Mathematics of Operations Research Vol. no. 15, pp. 311–341.MATHGoogle Scholar
- 151.Pang, J.S.: 1990, “Solution differentiability and continuation of Newton’s method for variational inequality problems over polyhedral sets”, Journal of Optimization Theory and Applications Vol. no. 66, pp. 121–135.MATHGoogle Scholar
- 152.Pang, J.S.: 1991, “A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity, and variational inequality problems”, Mathematical Programming Vol. no. 51, pp. 101–131.MATHGoogle Scholar
- 153.Pang, J.S.: 1992, “On local minima of nonlinear programs”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
- 154.Pang, J.S.: 1993, “Convergence of splitting and Newton methods for complementarity problems: an application of some sensitivity results”, Mathematical Programming Vol. no. 58, pp. 149–160.MATHGoogle Scholar
- 155.Pang, J.S.: 1993, “A degree-theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets”, Mathematical Programming, Series B, forthcoming.Google Scholar
- 156.Pang, J.S.: 1993, “Serial and parallel computations of Karush-Kuhn-Tucker points via non-smooth equations”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
- 157.Pang, J.S. and Chan, D.: 1982, “Iterative methods for variational and complementarity problems”, Mathematical Programming Vol. no. 24, pp. 284–313.MathSciNetMATHGoogle Scholar
- 158.Pang, J.S. and Gabriel, S.A.: 1993, “NE/SQP: A robust algorithm for the nonlinear complementarity problem”, Mathematical Programming, forthcoming.Google Scholar
- 159.Pang, J.S., Han, S.P., and Rangaraj, N.: 1991, “Minimization of locally Lipschitzian functions”, SIAM Journal of Optimization Vol. no. 1, pp. 57–82.MathSciNetMATHGoogle Scholar
- 160.Pang, J.S. and Qi, L.: 1993, “Nonsmooth equations: motivation and algorithms”, SIAM Journal on Optimization, Vol. no. 3, pp.Google Scholar
- 161.Pang, J.S. and Wang, Z.P.: 1990, “Embedding methods for variational inequality and nonlinear complementarity problems”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
- 162.Pang, J.S. and Yao, J.C.: 1992, “On a generalization of a normal map and equation”, manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore.Google Scholar
- 163.Pardalos, P.M. and Rosen, J.B.: 1987, Constrained Global Optimization: Algorithms and Applications, Springer Verlag, Berlin.Google Scholar
- 164.Pardalos, P.M. and Rosen, J.B.: 1988, “Global optimization approach to the linear complementarity problem”, SIAM Journal on Scientific and Statistical Computations Vol. no. 9, pp. 341–353.MathSciNetMATHGoogle Scholar
- 165.Qi, L.: 1993, “Convergence analysis of some algorithms for solving nonsmooth equations”, Mathematics of Operations Research Vol. no. 18, pp. 227–244.MATHGoogle Scholar
- 166.Qi, L. and Jiang, H.: 1992, “On the range sets of variational inequalities”, manuscript, Department of Applied Mathematics, University of New South Wales, Kensington, New South Wales, Australia.Google Scholar
- 167.Qi, L. and Sun, J.: 1993, “A nonsmooth version of Newton’s method”, Mathematical Programming Vol. no. 58, pp. 353–368.MathSciNetMATHGoogle Scholar
- 168.Qiu, Y. and Magnanti, T.: 1989, “Sensitivity analysis for variational inequalities defined on polyhedral sets”, Mathematics of Operations Research Vol. no. 14, pp. 410–332.MathSciNetMATHGoogle Scholar
- 169.Qiu, Y. and Magnanti, T.: 1992, “Sensitivity analysis for variational inequalities”, Mathematics of Operations Research Vol. no. 17, pp. 61–76.MathSciNetMATHGoogle Scholar
- 170.Ralph, D.: 1993, “Global convergence of damped Newton’s method for nonsmooth equations, via the path search”, Mathematics of Operations Research, forthcoming.Google Scholar
- 171.Reiman, M.I. and Williams, R.J.: 1988, “A boundary property of semi-martingale reflecting Brownian motions”, Probability Theory and Related Fields Vol. no. 77, pp. 87–97.MathSciNetMATHGoogle Scholar
- 172.Reinoza, A.: 1985, “The strong positivity conditions”, Mathematics of Operations Research Vol. no. 10, pp. 54–62.MathSciNetMATHGoogle Scholar
- 173.Robinson, S.M.: 1979, “Generalized equations and their applications, part I: basic theory”, Mathematical Programming Study Vol. no. 10, pp. 128–141.MATHGoogle Scholar
- 174.Robinson, S.M.: 1980, “Strongly regular generalized equations”, Mathematics of Operations Research Vol. no. 5, pp. 43–62.MATHGoogle Scholar
- 175.Robinson, S.M.: 1981, “Some continuity properties of polyhedral multifunction”, Mathematical Programming- Study Vol. no. 14, pp. 206–214.Google Scholar
- 176.Robinson, S.M.: 1982, “Generalized equations and their applications, part II: applications to nonlinear programming”, Mathematical Programming Study Vol. no. 19, pp. 200–221.MATHGoogle Scholar
- 177.Robinson, S.M.: 1983, “Generalized equations”, in A. Bachem, M. Grötschel, and B. Korte, eds., Mathematical Programming: The State of the Art, Springer Verlag, Berlin, pp. 346–367.Google Scholar
- 178.Robinson, S.M.: 1985, “Implicit B-differentiability in generalized equations”, Technical report #2854, Mathematics Research Center, University of Wisconsin, Madison.Google Scholar
- 179.Robinson, S.M.: 1987, “Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity”, Mathematical Programming Study, Vol. no. 30, pp. 45–66. Corrigenda, Mathematical Programming Vol. no. 49, pp. 143.Google Scholar
- 180.Robinson, S.M.: 1988, “Newton’s method for a class of nonsmooth functions”, manuscript, Department of Industrial Enginering, University of Wisconsin, Madison.Google Scholar
- 181.Robinson, S.M.: 1990, “Mathematical foundations of embedding methods for nonsmooth equations”, Mathematical Programming, Series B Vol. no. 48, pp. 221–230.MATHGoogle Scholar
- 190.Robinson, S.M.: 1990, “An implicit-function theorem for a class of nonsmooth functions”, Mathematics of Operations Research Vol. no. 16, pp. 292–309.Google Scholar
- 182.Robinson, S.M.: 1992, “Normal maps induced by linear transformations”, Mathematics of Operations Research Vol. no. 17, pp. 691–714.MATHGoogle Scholar
- 183.Robinson, S.M.: 1992, “Homeomorphism conditions for normal maps of polyhedra”, in A. Ioffe, M. Marcus, and S. Reich, eds., Optimization and Nonlinear Analysis, Longman, London, pp. 691–714.Google Scholar
- 184.Rockafellar, R.T.: 1970, Convex Analysis, Princeton University Press, Princeton.Google Scholar
- 185.Rockafellar, R.T.: 1987, “Linear-quadratic programming and optimal control”, SIAM Journal on Control and Optimization Vol. no. 25, pp. 781–814.MathSciNetMATHGoogle Scholar
- 186.Rockafellar, R.T.: 1989, “Proto-differentiability of set-valued mappings and its applications in optimization”, in H. Attouch, J.-P. Aubin, F.H. Clarke, and I. Ekeland, eds., Analyse Non Linéaire, Gauthier-Villars, Paris, pp. 449–482.Google Scholar
- 187.Rockafellar, R.T.: 1990, “Computational schemes for solving large-scale problems in extended linear-quadratic programming”, Mathematical Programming Vol. no. 48, pp. 447–474.MathSciNetMATHGoogle Scholar
- 188.Rockafellar, R.T.: 1990, “Computational schemes for solving large-scale problems in extended linear-quadratic programming”, Mathematical Programming Vol. no. 48, pp. 447–474.Google Scholar
- 189.Rockafellar, R.T. and Wets, R.J-B.: 1986, “A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming”, Mathematical Programming Studies Vol. no. 28, pp. 63–93.MathSciNetMATHGoogle Scholar
- 190.Rockafellar, R.T. and Wets, R.J-B.: 1987, “Linear-quadratic problems with stochastic penalties: the finite generation algorithm”, in Y. Ermoliev and R.J-B.Wets, eds., Numerical Techniques for Stochastic Optimization Problems, Springer-Verlag Lecture Notes in Control and Information Sciences Vol. no. 81, pp. 545–560.Google Scholar
- 200.Rockafellar, R.T. and Wets, R.J-B.: 1990, “Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time”, SIAM Journal on Control and Optimization Vol. no. 28, pp. 810–822.MathSciNetMATHGoogle Scholar
- 191.Scarf, H. (with the collaboration of T. Hansen): 1973, The Computation of Economic Equilibria, Yale University Press, New Haven.MATHGoogle Scholar
- 192.Stevens, S.N. and Lin, S.M.: 1981, “Analysis of piecewise linear resistive networks using complementarity pivot theory”, IEEE Transactions on Circuits and Systems Vol. no. CAS-28, pp. 429–441.MathSciNetGoogle Scholar
- 193.Stevens, S.N. and Lin, S.M.: 1981, “Analysis of piecewise linear resistive networks using complementarity pivot theory”, IEEE Transactions on Circuits and Systems Vol. no. CAS-28, pp. 429–441.Google Scholar
- 194.Sun., M.: 1987, “Singular control problems in bounded intervals”, Stochastics Vol. no. 21, pp. 303–344.MATHGoogle Scholar
- 195.Sun., M.: 1989, “Monotonicity of Mangasarian’s iterative algorithm for generalized linear complementarity problems”, Journal of Mathematical Analysis and Applications Vol. no. 144, pp. 474–485.MATHGoogle Scholar
- 196.Taji, K., Fukushima, M., and Ibaraki, T.: 1993, “A globally convergent Newton method for solving monotone variational inequalities”, Mathematical Programming Vol. no. 58, pp. 369–384.MathSciNetMATHGoogle Scholar
- 197.Thompson, G.L. and Thore, S.: 1991, “Economic disequilibrium by mathematical programming”, Journal of Optimization Theory and Applications Vol. no. 71, pp. 169–187.MathSciNetMATHGoogle Scholar
- 198.Thore, S., Nagurney, A., and Pan, J.: 1992, “Generalized goal programming and variational inequalities”, Operations Research Letters Vol. no. 12, pp. 217–226.MathSciNetMATHGoogle Scholar
- 199.Tin-Loi, F. and Pang, J.S.: 1993, “Elastoplastic analysis of structures with nonlinear hardening: a nonlinear complementarity approach”, Computer Methods in Applied Mechanics and Engineering, forthcoming.Google Scholar
- 200.Tobin, R.L.: 1986: “Sensitivity analysis for variational inequalities”, Journal of Optimization Theory and Applications Vol. no. 48, pp. 191–204.MathSciNetMATHGoogle Scholar
- 201.Todd, M.J.: 1976, Computation of Fixed Points and Applications, Lecture Notes in Economics and Mathematical Systems 124, Springer-Verlag, Heidelberg.Google Scholar
- 202.Todd, M.J. and Ye, Y.: 1990, “A centered projective algorithm for linear programming”, Mathematics of Operations Research Vol. no. 15, pp. 508–529.MathSciNetMATHGoogle Scholar
- 203.Vandenberghe, L., De Moor, B., and Vandewalle, J.: 1989, “The generalized linear complementarity problem problem applied to the complete analysis of resistive piecewise-linear circuits”, IEEE Transactions on Circuits and Systems Vol. no. 36, pp. 1382–1391.Google Scholar
- 204.Van Eijndhoven, J.T.L.: 1986, “Solving the linear complementarity problem in circuit simulation”, SIAM Journal on Control and optimization Vol. no. 24, pp. 1050–1062.MATHGoogle Scholar
- 205.Wright, S.: 1992, “An infeasible interior point algorithm for linear complementarity problems”, Technical report MCS-P331–1092, Mathematics and Computer Science Division, Argonne National Laboratory.Google Scholar
- 206.Wu, J.H.: 1992, “On descent simplicial decomposition methods for the monotone variational inequality problem with its application to the network equilibrium problem”, manuscript, Centre de Recherche sur les Transports, Université de Montréal, Montréal.Google Scholar
- 207.Wu, J.H., Florian, M., and Marcotte, P.: 1991, “A general descent framework for the mono-tone variational inequality problem”, Publication 723, Centre de Recherche sur les Transports, Université de Montréal, Montréal.Google Scholar
- 208.Yao, J.C.: 1990, “Generalized quasi variational inequality and implicit complementarity problems”, Ph.D. dissertation, Department of Operations Research, Stanford University, Stanford.Google Scholar
- 209.Yao, J.C.: 1991, “The generalized quasi-variational inequality problem with applications”, Journal of Mathematical Analysis and Applications Vol. no. 158, pp. 139–160.MATHGoogle Scholar
- 210.Yau, S.T. and Gao, Y.: 1992, “Obstacle problem for von K .rmân equations”, Advances in Applied Mathematics Vol. no. 13, pp. 123–141.MathSciNetMATHGoogle Scholar
- 211.Zarantonello, E.H.: 1971, “Projections on convex sets in Hilbert space and spectral theory”, in E.H. Zarantonello, ed., Contributions to Nonlinear Functional Analysis, Academic Press, New York, pp. 237–424.Google Scholar
- 212.Zhang, Y.: 1994, “On the convergence of an infeasible interior-point algorithm for linear programming and other problems”, SIAM Journal on Optimization Vol. no. 4, pp.Google Scholar
- 213.Zhu, C.Y.: 1992, “Modified proximal point algorithm for extended linear-quadratic programming”, Computational Optimization and Applications Vol. no. 2, pp. 182–205.Google Scholar
- 214.Zhu, C.Y. and Rockafellar, R.T.: 1993, “Primal-dual projected gradient algorithms for extended linear-quadratic progranuning”, SIAM Journal on Optimization Vol. no. 3, pp.Google Scholar
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