Abstract
This chapter is devoted to equilibrium problems and variational inequalities under generalized monotonicity assumptions on cost functions. We present basic existence and uniqueness results of solutions both for scalar and for vector problems. Relationships between generalized monotonicity properties of cost functions of these problems are also considered. Moreover, we describe basic approaches to construct iterative solution methods, including their convergence properties.
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Allen, G. Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. and Appl. 58, 1977, 1–10.
Ansari, Q.H. Vector equilibrium problems and vector variational inequalities, In: Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, F. Giannessi, ed., Kluwer Academic Publishers, Dordrecht-Boston-London, 2000, 1–15.
Ansari, Q.H., Konnov, I.V., and Yao, J.C. On generalized vector equilibrium problems, Nonl. Anal. 47, 2001, 543–554.
Ansari, Q.H., Oettli, W., and Schlãger, D. A generalization of vectorial equilibria, Math. Meth. Oper. Res. 46, 1997, 147–152.
Antipin, A.S. On convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence, Comp. Maths. Math. Phys. 35, 1995, 539–551.
Arrow, K.J. and Hahn, F.H. General competitive analysis, Holden Day, San Francisco, 1972.
Aubin, J.-P. Optima and equilibria, Springer-Verlag, Berlin, 1998.
Aussel, D., Corvellec, J.-N., and Lassonde, M. Subdifferential characterization of quasiconvexity and convexity, J. Convex Analysis 1, 1994, 195–201.
Avriel, M., Diewert, W.E., Schaible, S., and Zang, I. Generalized convexity, Plenum Press, New York, 1988.
Baiocchi, C. and Capelo, A. Variational and quasivariational inequalities. Applications to free boundary problems, John Wiley and Sons, New York, 1984.
Bakushinskii, A.B. and Goncharskii, A.V. Ill-posed problems: Theory and applications, Kluwer Academic Publishers, Dordrecht, 1994.
Belen’kii, V.Z. and Volkonskii, V.A., Editors, Iterative methods in game theory and programming, Nauka, Moscow, 1974. (in Russian)
Berge, C. Topological spaces, Oliver and Boyd, Edinburgh, 1963.
Bertsekas, D.P. Constrained optimization and Lagrange multiplier methods, Academic Press, New York, 1982.
Bianchi, M. Pseudo P-monotone operators and variational inequalities, Research Report No. 6, Istituto di Econometrica e Matematica per le Decisioni Economiche, Universita Cattolica del Sacro Cuore, Milan, 1993.
Bianchi, M., Hadjisavvas, N., and Schaible, S. Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl. 92, 1997, 527–542.
Bianchi, M. and Pini, R. A note on equilibrium problems for KKM-maps, Quaderno No. 37, Istituto di Econometrica e Matematica per le Applicazioni Economiche, Finanziare ed Attuari, Milan, 2000.
Bianchi, M. and Schaible, S. Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl. 90, 1996, 31–43.
Billups, S.C. and Ferris, M.C. QPCOMP: A quadratic programming based solver for mixed complementarity problems, Math. Progr. 76, 1997, 533–562.
Blackwell, D. An analogue of the minimax theorem for vector payoffs, Pacific J. Math. 6, 1956, 1–8.
Blum, E. and Oettli, W. Variational principles for equilibrium problems, In: Parametric Optimization and Related Topics III, J. Guddat, H.Th. Jongen, B. Kummer, and F.Nozicka eds., Peter Lang Verlag, Frankfurt am Main, 1993, 79–88.
Blum, E. and Oettli, W. From optimization and variational inequalities to equilibrium problems, The Math. Student 63, 1994, 127–149.
Borwein, J.M. and Lewis, A.S. Partially finite convex programming, Part I: Quasi relative interiors in duality theory, Math. Progr. 57, 1992, 15–48.
Brézis, H., Nirenberg, L., and Stampacchia, G. A remark on Ky Fan’s minimax principle, Boll. Unione Mat. Ital. 6, 1972, 293–300.
Browder, F.E. Multivalued monotone nonlinear mappings and duality mappings in Banach spaces, Trans. Amer. Math. Soc. 71, 1965, 780–785.
Bruck, R. On weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61, 1977, 159–164.
Cambini, R. and Komlósi, S. On the scalarization of pseudoconvexity and pseudomonotonicity concepts for vector-valued functions, in: Generalized Convexity, Generalized Monotonicity, J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht-Boston-London, 1998, 277–290.
Cambini, R. and Komlósi, S. On polar generalized monotonicity in vector optimization, Optimization 47, 2000, 111–121.
Censor, Y., Lusem, A.N., and Zenios, S. An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Progr. 81, 1999, 373–400.
Chadli, O., Chbani, Z., and Riahi, H. Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities, J. Optim. Theory Appl. 105, 2000, 299–323.
Chen, G.Y. Existence of solution for a vector variational inequality: An extension of the Hartmann-Stampacchia theorem, J. Optim. Theory Appl. 74, 1992, 445–456.
Chen, G.Y. and Cheng, G.M. Vector variational inequality and vector optimization, Lecture Notes in Economics and Mathematical Systems 258, Springer-Verlag, Heidelberg, 1987, 408–416.
Chen, G. Y. and Craven, B.D. A vector variational inequality and optimization over an efficient set, Zeit. Oper. Res. 3, 1990, 1–12.
Chen, G. Y. and Craven, B.D. Existence and continuity for vector optimization, J. Optim. Theory Appl. 81, 1994, 459–468.
Chen, G.Y. and Yang, X.Q., Vector complementarity problem and its equivalences with weak minimal element in ordered spaces, J. Math. Anal. Appl. 153, 1990, 136–158.
Clarke, F.H. Optimization and nonsmooth analysis, John Wiley and Sons, New York, 1983.
Crouzeix, J.-P. Pseudomonotone variational inequality problems: Existence of solutions, Math. Progr. 78, 1997, 305–314.
Crouzeix, J.-P. Characterization of generalized convexity and generalized monotonicity: A survey, in: Generalized Convexity, Generalized Monotonicity, J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht-Boston-London, 1998, 237–256.
Crouzeix, J.-P. and Ferland, J. A. Criteria for differentiate generalized monotone maps, Math. Progr. 75, 1996, 399–406.
Crouzeix, J.-P. and Hassouni, A. Quasimonotonicity of separable operators and monotonicity indices, SIAM J. Optim. 4, 1994, 649–658.
Crouzeix, J.-P. and Hassouni, A. Generalized monotonicity of a separable product of operators: The multivalued case, Set-Valued Analysis 3, 1995, 351–373.
Crouzeix, J.-P., Hassouni, A., Lahlou, A., and Schaible, S. Positive definite matrices, generarized monotonicity and linear complementarity problems, SIAM J. Matrix Anal. Appl. 22, 2000, 66–85.
Crouzeix, J.-P., Marcotte, P., and Zhu, D.L. Conditions ensuring the applicability of cutting plane methods for solving variational inequalities, Math. Progr. 88, 2000, 521–539.
Crouzeix, J.-P., and Schaible, S. Generalized monotone affine maps, SIAM J. Matrix Anal. Appl. 17, 1996, 992–997.
Daniilidis, A. and Hadjisavvas, N. Variational inequalities with quasimonotone multivalued operators, Working Paper, Department of Mathematics, University of the Aegean, Samos, Greece, March 1995.
Daniilidis, A. and Hadjisavvas, N. Existence theorems for vector variational inequalities, Bull. Austral. Math. Soc. 54, 1996, 473–481.
Daniilidis, A. and Hadjisavvas, N. Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions, J. Optim. Theory Appl. 102, 1999, 525–536.
Daniilidis, A. and Hadjisavvas, N. Coercivity conditions and variational inequalities, Math. Progr. 86, 1999, 433–438.
Daniilidis, A. and Hadjisavvas, N. On the subdifferentials of quasiconvex and pseudoconvex functions and cyclic monotonicity, J. Math. Anal. Appl. 237, 1999, 30–42.
Daniilidis, A. and Hadjisavvas, N. On generalized cyclically monotone operators and proper quasimonotonicity, Optimization 47, 2000, 123–135.
Dem’yanov, V.F. and Rubinov, A.M. Approximate methods for solving extremum problems, Leningrad University Press, Leningrade, 1968 (in Russian; Engl. transl. in Elsevier Science B.V., Amsterdam, 1970).
Denault, M. and Goffin, J.-L. On a primal-dual analytic center cutting plane method for variational inequalities, Comp. Optim. Appl.. 12, 1999, 127–155.
Ding, X.P. Browder-Hartmann-Stampacchia variational inequalities, J. Math. Anal. Appl. 173, 1993, 577–587.
Ding, X.P. and Tarafdar, E. Generalized nonlinear variational inequalities with non monotone set-valued mappings, Appl. Math. Lett. 7, 1994, 5–11.
Ding, X.P. and Tarafdar, E. Generalized variational-like inequalities with pseudomonotone set-valued mappings, Arch. Math. 74, 2000, 302–313.
Duvaut, G. and Lions, J.-L. Les inéquations en mechanique et physique, Dunod, Paris, 1972.
Fan Ky, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142, 1961, 305–310.
Fan Ky, A minimax inequality and applications, In: Inequalities. III, O. Shisha, ed., Academic Press, New York, 1972, 103–113.
Fan Ky, Some properties of convex sets related to fixed point theorems, Math. Ann. 266, 1984, 519–537.
Fang, S.C. and Petersen, E.L. Generalized variational inequalities, J. Optim. Theory Appl.. 38, 1982, 363–383.
Farquharson, R. Sur une généralisation de la notion d’équilibrium, Compt. Rend. Acad. Sci. Paris 240, 1955, 46–48.
Ferris, M.C. and Pang, J.-S. Engineering and economic applications of complementarity problems, SIAM Review 39, 1997, 669–713.
Fichera, G. Problemi elastostatici con vincoli unilaterali; il problema di Signorini con ambigue al contorno, Acad. Naz. Lincei. Mem. 8, 1964, 91–140.
Fukushima, M. Merit functions for variational inequality and complementarity problems, In: Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi. Plenum Press, New York, 1996, 155–170.
Giannessi, F. Theorems of alternative, quadratic programs and complementarity problems, in: Variational Inequalities and Complementarity Problems, R.W. Cottle, F. Giannessi, and J.-L. Lions, eds., John Wiley and Sons, Chichester, 1980, 151–186.
Giannessi, F. On Minty variational principle, in: New Trends in Mathematical Programming, F. Giannessi, S. Komlósi, and T. Rapcsák, eds., Kluwer Academic Publishers, Dordrecht, 1998, 93–99.
Giannessi, F., Editor, Vector variational inequalities and vector equilibria. Mathematical theories, Kluwer Academic Publishers, Dordrecht-Boston-London, 2000.
Giannessi, F. and Maugeri, A., Editors, Variational inequalities and network equilibrium problems, Plenum Press, New York, 1995.
Giannessi, F. and Pellegrini, L., Image space analysis for vector optimization and variational inequalities. Scalarization, In: Proceedings of Combinatorial and Global Optimization Conference, Chania, May 1998, P. Pardalos, ed., Kluwer Academic Publishers, Dordrecht-Boston-London, to appear.
Gill, P.E., Murray, W., and Wright, M.H. Practical optimization, Academic Press, New York, 1981.
Glowinski, R., Lions, J.-L., and Trémolières, R. Analyse numerique des inéquations variationnelles, Dunod, Paris, 1976.
Goffin, J.-L., Marcotte, P., and Zhu, D.L. An analytic center cutting plane method for pseudomonotone variational inequalities, Oper. Res. Lett. 20, 1999, 1–6.
Goh, C.J. and Yang, X.Q. Vector equilibrium problems and vector optimization, Eur. J. Oper. Res. 116, 1999, 615–628.
Gol’shtein, E.G., Nemirovskii, A.S., and Nesterov, Yu.E. Level method, its extensions and applications, Ekon. Mat. Met. 31, 1995, 164–181. (in Russian)
Gol’shtein, E.G. and Tret’yakov, N.V. Modified Lagrangians and monotone maps in optimization, John Wiley and Sons, New York, 1996.
Gowda, M.S. Pseudomonotone and copositive star matrices, Linear Algebra Appl. 113, 1989, 107–118.
Gowda, M.S. Affine pseudomonotone mappings and the linear complementarity problem, SIAM J. Matr. Anal. Appl. 11, 1990, 373–380.
Granas, A. Sur quelques méthodes topologiques en analyse convexe, in: Méthodes Topologiques en Analyse Convexe, A. Granas, ed., Les Presses de l’Université de Montreal, Montreal, 1990, 151–186.
Hadjisavvas, N., Kravvaritis, D., Pantelidis, G., and Polyrakis, I. Nonlinear monotone operators with values in L(X, Y), J. Math. Anal. Appl. 140, 1989, 83–94.
Hadjisavvas, N. and Schaible, S. On strong pseudomonotonicity and (semi) strict quasimonotonicity, J. Optim. Theory Appl. 79, 1993, 139–155.
Hadjisavvas, N. and Schaible, S. Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl. 90, 1996, 95–111.
Hadjisavvas, N. and Schaible, S. From scalar to vector equilibrium problems in the quasimonotone case, J. Optim. Theory Appl. 96, 1998, 297–309.
Hadjisavvas, N. and Schaible, S. Quasimonotonicity and pseudomonotonicity in variational inequalities and equilibrium problems, in: Generalized Convexity, Generalized Monotonicity, J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht-Boston-London, 1998, 257–275.
Harker, P.T. and Pang, J.-S. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Progr. 48, 1990, 161–220.
Hartmann, P. and Stampacchia, G. On some nonlinear elliptic differential functional equations, Acta Mathematica 115, 1966, 153–188.
Hassouni, A. Quasimonotone multifunctions; applications to optimality conditions in quasiconvex programming, Num. Funct. Anal, and Optim. 13, 1992, 267–275.
He, B. A class of projection and contraction methods for monotone variational inequalities, Appl. Math, and Optim. 35, 1997, 69–76.
Jeyakumar, V., Oettli, W., and Natividad, M. A solvability theorem for a class of quasiconvex mappings with applications to optimization, J. Math. Anal. Appl. 179, 1993, 537–546.
John, R. Variational inequalities and pseudomonotone functions: some characterizations, in: Generalized Convexity, Generalized Monotonicity, J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht-Boston-London, 1998, 291–301.
John, R. A note on Minty variational inequalities and generalized monotonicity, in: Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas, J.-E. Martinez-Legaz, and M. Penot, eds., Lecture Notes in Economics and Mathematical Systems 502, Springer-Verlag, Berlin-Heidelberg-New York, 2001, 240–246.
Kachurovskii, R.I. On monotone operators and convex functionals, Usp. Matem. Nauk 15, 1960, 213–215.
Kantorovich, L.V. and Akilov, G.P. Functional analysis in normed spaces, Fizmatgiz, Moscow, 1959. (in Russian)
Kanzow, C. and Qi, H.-D. A QP-free constrained Newton-type method for variational inequality problems, Math. Progr. 85, 1999, 81–106.
Karamardian, S. Generalized complementarity problems, J. Optim. Theory Appl. 8, 1971, 161–167.
Karamardian, S. An existence theorem for the complementarity problem, J. Optim. Theory Appl. 19, 1976, 227–232.
Karamardian, S. Complementarity over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18, 1976, 445–454.
Karamardian, S. and Schaible, S. Seven kinds of monotone maps, J. Optim. Theory Appl. 66, 1990, 37–46.
Karamardian, S., Schaible, S., and Crouzeix, J.-P. Characterizations of generalized monotone maps, J. Optim. Theory Appl. 76, 1993, 399–413.
Kazmi, K.R. Existence of solutions for vector optimization, Appl. Math. Letters 9, 1996, 19–22.
Kazmi, K.R. Some remarks on vector optimization problems, J. Optim. Theory Appl. 96, 1998, 133–138.
Kazmi, K.R. Existence of solutions for vector saddle points problems, In: Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, F. Giannessi, ed., Kluwer Academic Publishers, Dordrecht-Boston-London, 2000, 267–275.
Kazmi, K.R. On vector equilibrium problem, Proc. Indian Acad. Sci. (Math. Sci.) 110, 2000, 213–223.
Khobotov, E.N. Modification of the extragradient method for solving variational inequalities and certain optimization problems, USSR Comp. Maths. Math. Phys. 27, 1987, 120–127.
Kinderlehrer, D. and Stampacchia, G. An introduction to variational inequalities and their applications, Academic Press, New York, 1980.
Kiwiel, K.C. Proximal level bandle methods for convex nondifferentiale optimization, saddle point problems and variational inequalities, Math. Progr. 69, 1995, 89–109.
Knaster, B., Kuratowski, C., and Mazurkiewicz, S., Ein Beweis des Fixpunktsatzes für N Dimensionale Simplexe, Fund. Math. 14, 1929, 132–137.
Kneser, H. Sur le théorèms fondamental de la théorie des jeux, C. R. Acad. Sci. Paris 234, 1952, 2418–2420.
Komlósi, S. Generalized monotonicity in nonsmooth analysis, In: Generalized Convexity, S. Komlósi, T. Rapcsák, and S. Schaible, eds., Springer-Verlag, Heidelberg, 1994, 263–275.
Komlósi, S. On the Stampacchia and Minty variational inequalities, in: Generalized Convexity and Optimization in Economic and Financial Decisions, G. Giorgi and F. Rossi, eds., Pitagora Editrice, Bologna, 1999, 231–260.
Konnov, I.V. Combined relaxation methods for finding equilibrium points and solving related problems, Russ. Math. (Iz. VUZ) 37, 1993, no.2, 44–51.
Konnov, I.V. A combined method for variational inequalities, News-Lett. Math. Progr. Assoc. 4, 1993, 64. (in Russian)
Konnov, I.V. On combined relaxation method’s convergence rates, Russ. Math. (Iz. VUZ) 37, 1993, no. 12, 89–92.
Konnov, I.V. Combined relaxation methods for solving vector equilibrium problems, Russ. Math. (Iz. VUZ) 39, 1995, no. 12, 51–59.
Konnov, I.V. A general approach to finding stationary points and the solution of related problems, Comp. Maths. Math. Phys. 36, 1996, 585–593.
Konnov, I.V. A class of combined iterative methods for solving variational inequalities, J. Optim. Theory Appl. 94, 1997, 677–693.
Konnov, I.V. On systems of variational inequalities, Russ. Math. (Iz. VUZ) 41, 1997, no. 12, 77–86.
Konnov, I.V. Methods for solving finite-dimensional variational inequalities, DAS, Kazan, 1998. (in Russian)
Konnov, I.V. A combined relaxation method for variational inequalities with nonlinear constraints, Math. Progr. 80, 1998, 239–252.
Konnov, I.V. Accelerating the convergence rate of a combined relaxational method, Comp. Maths. Math. Phys. 38, 1998, 49–56.
Konnov, I.V. On the convergence of combined relaxation methods for variational inequalities, Optim. Meth. Software 80, 1998, 239–252.
Konnov, I.V. On quasimonotone variational inequalities, J. Optim. Theory Appl. 99, 1998, 165–181.
Konnov, I.V. Combined relaxation methods for variational inequalities, Lecture Notes in Economics and Mathematical Systems 495, Springer-Verlag, Berlin-Heidelberg-New York, 2001.
Konnov, I.V. On vector equilibrium and vector variational inequality problems, in: Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas, J.-E. Martinez-Legaz, and M. Penot, eds., Lecture Notes in Economics and Mathematical Systems 502, Springer-Verlag, Berlin-Heidelberg-New York, 2001, 247–263.
Konnov, I.V. Relatively monotone variational inequalities over product sets, Oper. Res. Letters 28, 2001, 21–26.
Konnov, I.V. and Schaible, S. Duality for equilibrium problems under generalized monotonicity, J. Optim. Theory Appl. 104, 2000, 395–408.
Konnov, I.V. and Yao, J.C. On the generalized vector variational inequality problem, J. Math. Anal. Appl. 206, 1997, 42–58.
Konnov, I.V. and Yao, J.C. Existence of solutions for generalized vector equilibrium problems, J. Math. Anal. Appl. 233, 1999, 328–335.
Korpelevich, G.M. Extragradient method for finding saddle points and other problems, Matecon 12, 1976, 747–756.
Lee, G.M., Kim, D.S., Lee, B.S., and Cho, S.J. Generalized vector variational inequalities and fuzzy extension, Appl. Math. Letters 6, 1993, 47–51.
Lee, G.M., Kim, D.S., Lee, B.S., and Yen, N.D. Vector variational inequalities as a tool for studying vector optimization problems, Nonlin. Anal. 34, 1998, 745–765.
Lee, G.M. and Kum, S. Vector variational inequalities in a topological vector space, In: Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, F. Giannessi, ed., Kluwer Academic Publishers, Dordrecht-Boston-London, 2000, 307–320.
Lemaréchal, C., Nemirovskii, A., Nesterov, Y. New variants of bundle methods Math. Progr. 69, 1995, 111–147.
Lemke, C.E. Bimatrix equilibrium points and mathematical programming, Manag. Sci. 11, 1965, 681–689.
Levin, A.L. On an algorithm for the minimization of convex functions, Soviet Math. Doklady 6, 1965, 286–290.
Levin, V.L. Quasiconvex functions and quasimonotone operators, J. Convex Analysis 2, 1995, 167–172.
Lin, G.-H. and Xia, Z.-Q. Some convergence results for pseudomonotone variational inequalities, Pure Math. Appl. 10, 1999, 173–182.
Lin, K.L., Yang, D.P., and Yao, J.C. Generalized vector variational inequalities, J. Optim. Theory Appl. 92, 1997, 117–125.
Lin, L.-J. and Park, S. On some generalized quasi-equilibrium problems, J. Math. Anal. Appl. 224, 1998, 167–181.
Lions, J.-L. and Stampacchia, G. Variational inequalities, Comm. Pure Appl. Math. 20, 1967, 493–519.
Luc, D.T. Theory of vector optimization, Lecture Notes in Mathematics 543, Springer-Verlag, Berlin, 1989.
Luc, D.T. Characterizations of quasiconvex functions, Bull. Austral. Math. Soc. 48, 1993, 393–406.
Luc, D.T. Existence results for densely pseudomonotone variational inequalities, J. Math. Anal. Appl. 254, 2001, 291–308.
Luc, D.T. and Schaible, S. Generalized monotone nonsmooth maps, J. Convex Analysis 3, 1996, 195–205.
Magnanti, T.L. and Perakis, G. A unifying geometric framework and complexity analysis for variational inequalities, Math. Progr. 71, 1995, 327–352.
Marcotte, P. and Zhu, D.L. A cutting plane method for quasimonotone variational inequalities, Comp. Math. Appl. 20, 2001, 317–324.
Martinet, B. Regularization d’inéquations variationnelles par approximations successives, Rev. Fr. Inf. Rech. Oper. 4, 1970, 154–159.
Martos, B. Nonlinear programming. Theory and methods, Akadémiai Kiado, Budapest, 1975.
Minty, G. On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc. 73, 1967, 315–321.
Mosco, U. Implicit variational problems and quasivariational inequalities, In: Nonlinear Operators and Calculus of Variations, J.P. Gossez et al., eds., Lecture Notes in Mathematics 543, Springer-Verlag, Berlin, 1976, 83–156.
Nagurney, A. Network economics: A variational inequality approach, Kluwer Academic Publishers, Dordrecht, 1993.
Nemirovskii, A.S. Effective iterative methods for solving equations with monotone operators, Ekon. Matem. Met. (Matecon) 17, 1981, 344–359.
Nemirovskii, A.S. and Yudin, D.B. Problem complexity and method efficiency in optimization, Nauka, Moscow, 1979. (in Russian; Engl. transl. in John Wiley and Sons, New York, 1983).
von Neumann, J. and Morgenstern, O. J Game theory and economic behaviour, Princeton University Press, Princeton, 1953.
Nikaido, H. Convex structures and economic theory, Academic Press, New York, 1968.
Nikaido, H. and Isoda, K. Note on noncooperative convex games, Pacific J. Math. 5, 1955, 807–815.
Nirenberg, L. Topics in nonlinear functional analysis, New York University Press, New York, 1974.
Oettli, W. A remark on vector-valued equilibria and generalized monotonicity, Acta Mathem. Vietnam. 22, 1997, 213–221.
Oettli, W. and Schlãger, D. Generalized vectorial equilibria and generalized monotonicity, in Functional Analysis with Current Applications in Science, Technology and Industry, M. Brokate and A.H. Siddiqi, eds., Pitman Research Notes in Mathematics Series 377, Longman Publishing Co., Essex, 1998, 145–154.
Ortega, J.M. and Rheinboldt, W.C. Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970.
Patriksson, M. Nonlinear programming and variational inequality problems: A unified approach, Kluwer Academic Publishers, Dordrecht, 1999.
Pellegrini, L. Vector formulation of some economic equilibrium problems, in: Generalized Convexity and Optimization in Economic and Financial Decisions, G. Giorgi and F. Rossi, eds., Pitagora Editrice, Bologna, 1999, 323–331.
Penot, J.-P. and Quang, P.H. Generalized convexity of functions and generalized monotonicity of set-valued maps, J. Optim. Theory Appl. 92, 1997, 343–356.
Podinovskii, V.V. and Nogin, V.D. Pareto-optimal solutions of multicriteria problems, Nauka, Moscow, 1982 (in Russian).
Rockafellar, R.T. Convex analysis, Princeton University Press, Princeton, 1970.
Rockafellar, R.T. Monotone operators and the proximal point algorithm, SIAM J. Contr. Optim. 14, 1976, 877–898.
Rockafellar, R.T. The theory of subgradients and its applications to problems of optimization, Heldermann-Verlag, Berlin, 1981.
Rosen, J.B. Existence and uniqueness of equilibrium points for concave n-person games, Econometrica 33, 1965, 520–534.
Sach, P.H. Characterization of scalar quasiconvexity and convexity of locally Lipschitz vector valued maps, Optimization 46, 1999, 283–310.
Saigal, R. Extension of the generalized complementarity problem, Math. Oper. Res. 1, 1976, 260–266.
Schaible, S. Generalized monotone maps, In: Proceedings of a Conference Held at “G. Stampacchia International School of Mathematics”, F. Giannessi, ed., Gordon and Breach Science Publishers, Amsterdam, 1992, 392–408.
Schaible, S. Generalized monotonicity-concepts and uses, in: Variational Inequalities and Network Equilibrium Problems, F. Giannessi and A. Maugeri, eds., Plenum Press, New York, 1995, 289–299.
Schaible, S. Criteria for generalized monotonicity, in: New Trends in Mathematical Programming, F. Giannessi, S. Komlósi, and T. Rapcsák, eds., Kluwer Academic Publishers, Dordrecht, 1998, 277–288.
Shih, M.H. and Tan, K.K. Browder-Hartmann-Stampacchia variational inequalities for multivalued monotone operators, J. Math. Anal. Appl. 134, 1988, 431–440.
Shor, N.Z. Cut-off method with space dilation in convex programming problems, Cybernetics 13, 1977, 94–96.
Shor, N.Z. New development trends in nonsmooth optimization methods, Cybernetics 6, 1977, 87–91.
Shor, N.Z. Minimization methods for non-differentiate functions, Naukova Dumka, Kiev, 1979. (in Russian; Engl. transl. in Springer-Verlag, Berlin, 1985).
Solodov, M.V. and Tseng, P. Modified projection-type methods for monotone variational inequalities, SIAM J. Contr. Optim. 34, 1996, 1814–1830.
Stampacchia, G. Formes bilinéaires coercitives sur le ensembles convexes, C. R. Acad. Sci. Paris 258, 1964, 4413–4416.
Sun, D. A new step-size skill for solving a class of nonlinear projection equations, J. Comp. Math. 13, 1995, 357–368.
Sun, D. A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory Appl. 91, 1996, 123–140.
Tanaka, T. Cone convexity of vector-valued functions, Sci. Rep. Hirosaki Univ. 37, 1990, 170–177.
Tanaka, T. Generalized quasiconvexities, cone saddle points, and minimax theorems for vector-valued functions, J. Optim. Theory Appl. 81, 1994, 355–377.
Tanaka, T. Cone quasiconvexity of vector-valued functions, Sci. Rep. Hirosaki Univ. 42, 1995, 157–163.
Tanino, T. and Sawaragi, Y. Duality theory in multiobjective optimization, J. Optim. Theory Appl. 27, 1979, 509–529.
Tikhonov, A.N. and Arsenin, A.V. Solution of ill-posed problems, John Wiley and Sons, New York, 1977.
Tseng, P. On linear convergence of iterative methods for the variational inequality problem, J. Comp. Appl. Math. 60, 1995, 237–252.
Uryas’yev, S.P. Adaptive algorithms of stochastic optimization and game theory, Nauka, Moscow, 1990. (in Russian)
Yang, X.Q. Vector variational inequality and its duality, Nonlin. Anal., Theory, Methods and Appl. 21, 1993, 869–877.
Yang, X.Q. Vector complementarity and minimal element problems, J. Optim. Theory Appl. 77, 1993, 483–495.
Yang, X.Q. and Goh, C.J. On vector variational inequalities: Applications to vector equilibria, J. Optim. Theory Appl. 95, 1997, 431–443.
Yao, J.C. Variational inequalities with generalized monotone operators, Math. Oper. Res. 19, 1994, 691–705.
Yao, J.C. Multi-valued variational inequalities with K-pseudomonotone operators, J. Optim. Theory Appl. 83, 1994, 391–403.
Yen, N.D. and Lee, G.M. On monotone and strongly monotone vector variational inequalities, in: Generalized Convexity and Optimization in Economic and Financial Decisions, G. Giorgi and F. Rossi, eds., Pitagora Editrice, Bologna, 1999, 347–358.
Yu, S.J. and Yao, J.C. On vector variational inequalities, J. Optim. Theory Appl. 89, 1996, 749–769.
Yuan, G.X.Z. The study of minimax inequalities and applications to economies and variational inequalities, Memoires of the American Mathematical Society 132, Number 625, American Mathematical Society, Providence, 1998.
Yudin, D.B. and Nemirovskii, A.S. Informational complexity and efficient methods for the solution of convex extremal problems, I, II, Matecon 13, 1977, 25–45; 550–559.
Zarantonello, E.H. Projections on convex sets in Hilbert space and spectral theory, in: Contributions to Nonlinear Functional Analysis, E.H. Zarantonello, ed., Academic Press, New York-London, 1971, 237–424.
Zhou, J.X., and Chen, G. On diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl. 132, 1988, 213–225.
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Konnov, I. (2005). Generalized Monotone Equilibrium Problems and Variational Inequalities. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds) Handbook of Generalized Convexity and Generalized Monotonicity. Nonconvex Optimization and Its Applications, vol 76. Springer, New York, NY. https://doi.org/10.1007/0-387-23393-8_13
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