Summary
The paper is devoted to the combined relaxation approach to constructing solution methods for variational inequalities. We describe the basic idea of this approach and implementable methods both for single-valued and for multi-valued problems. All the combined relaxation methods are convergent under very mild assumptions. This is the case if there exists a solution to the dual formulation of the variational inequality problem. In general, these methods attain a linear rate of convergence. Several classes of applications are also described.
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References
Agmon S (1954): The relaxation method for linear inequalities. Canadian Journal of Mathematics 6: 382–392
Arrow KJ, Hahn FH (1972): General competitive analysis. Holden Day, San Francisco
Arrow KJ, Hurwicz L (1960): Competitive stability under weak gross substitutability: The “Euclidean distance” approach, International Economic Review 1: 38–49
Baiocchi C, Capelo A (1984): Variational and quasivariational inequalities. Applications to free boundary problems. John Wiley and Sons, New York
Belen’kii VZ, Volkonskii VA (eds) (1974): Iterative methods in game theory and programming. Nauka, Moscow (in Russian)
Bianchi M, Schaible S (1996): Generalized monotone bifunctions and equilibrium problems. Journal of Optimization Theory and Applications 90: 31–43
Cimmino G (1938): Calcolo approsimato per le soluzioni dei sistemi de equazioni lineari. La Ricerca Scientifica XVI 1, 326–333
Eremin II (1965): The relaxation method of solving systems of inequalities with convex functions on the left-hand sides. Soviet Mathematics Doklady 6: 219–222
Facchinei F, Pang JS (2003): Finite-dimensional variational inequalities and complementarity problems. Springer-Verlag, Berlin
Fishburn PC (1970): Utility theory for decision making. Wiley, New York
Hadjisavvas N, Komlósi S, Schaible S (eds) (2005): Handbook of generalized convexity and generalized monotonicity. “Nonconvex optimization and applications”, Vol.76. Springer, New York
Kaczmarz S (1937): Angenäherte Auflösung von Systemen linearer Gleichungen. Bulletin Internationel de l’Académie Polonaise des Sciences et des Lettres. Classe des Sciences Mathématiques et Naturelles. Séries A 60: 596–599
Karamardian S (1976): Complementarity over cones with monotone and pseudomonotone maps. Journal of Optimization Theory and Applications 18: 445–454
Komlósi S (1999): On the Stampacchia and Minty variational inequalities. In: Giorgi G, Rossi F (eds) Generalized convexity and optimization in economic and financial decisions. Pitagora Editrice, Bologna, 231–260
Konnov IV (1982): A subgradient method of successive relaxation for solving optimization problems. Preprint VINITI No. 531-83, Kazan University, Kazan (in Russian)
Konnov IV (1984): A method of the conjugate subgradient type for minimization of functionals. Issledovaniya po Prikladnoi Matematike 12: 59–62 (Engl. transl. in Journal of Soviet Mathematics 45: 1026–1029, 1989).
Konnov IV (1993): Combined relaxation methods for finding equilibrium points and solving related problems, Russian Mathematics (Iz. VUZ) 37(2): 44–51
Konnov IV (1993): On combined relaxation method’s convergence rates. Russian Mathematics (Iz. VUZ) 37(12): 89–92
Konnov IV (1994): Applications of the combined relaxation method to finding equilibrium points of a quasi-convex-concave function. Russian Mathematics (Iz. VUZ) 38(12): 66–71
Konnov IV (1996): A general approach to finding stationary points and the solution of related problems. Computational Mathematics and Mathematical Physics 36: 585–593
Konnov IV (1997): A class of combined iterative methods for solving variational inequalities. Journal of Optimization Theory and Applications 94, 677–693
Konnov IV (1997): Combined relaxation methods having a linear rate of convergence. News-Letter of the Mathematical Programming Association 7: 135–137 (in Russian)
Konnov IV (1998): A combined relaxation method for variational inequalities with nonlinear constraints. Mathematical Programming 80: 239–252
Konnov IV (1998): On the convergence of combined relaxation methods for variational inequalities. Optimization Methods and Software 9: 77–92
Konnov IV (1998): An inexact combined relaxation method for multivalued inclusions. Russian Mathematics (Iz. VUZ) 42(12): 55–59
Konnov IV (1999): A combined method for variational inequalities with monotone operators. Computational Mathematics and Mathematical Physics 39:1051–1056
Konnov IV (1999): Combined relaxation methods for variational inequality problems over product sets. Lobachevskii Journal of Mathematics 2: 3–9
Konnov IV (1999): Combined relaxation method for decomposable variational inequalities. Optimization Methods and Software 10: 711–728
Konnov IV (2001): Combined relaxation methods for variational inequalities. Springer-Verlag, Berlin
Konnov IV (2001): Combined relaxation method for extended variational inequalities. Russian Mathematics (Iz. VUZ) 45(12): 46–54
Konnov IV (2002): A combined relaxation method for nonlinear variational inequalities. Optimization Methods and Software 17: 271–292
Konnov IV (2002): A class of combined relaxation methods for decomposable variational inequalities. Optimization 51: 109–125
Konnov IV (2002): A combined relaxation method for a class of nonlinear variational inequalities. Optimization 51: 127–143
Minty G (1967): On the generalization of a direct method of the calculus of variations. Bulletin of the American Mathematical Society 73: 315–321
Motzkin TS, Schoenberg IJ (1954): The relaxation method for linear inequalities. Canadian Journal of Mathematics 6: 393–404
Nagurney A (1999): Network economics: a variational inequality approach. Kluwer Academic Publishers, Dordrecht
Nikaido H (1968): Convex structures and economic theory. Academic Press, New York
Nikaido H, Isoda K (1955): Note on noncooperative convex games. Pacific Journal of Mathematics 5: 807–815
Polterowich VM, Spivak VA (1983): Gross substitutability of point-to-set correspondences. Journal of Mathematical Economics 11: 117–140
Polyak BT (1969): Minimization of unsmooth functionals. USSR Computational Mathematics and Mathematical Physics 9: 14–29
Rosen JB (1965): Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33: 520–534
Shafer W (1974): The non-transitive consumer. Econometrica 42: 913–919
Shih MH, Tan KK (1988): Browder-Hartmann-Stampacchia variational inequalities for multivalued monotone operators. Journal of Mathematical Analysis and Applications 134: 431–440
Sen AK (1970): Collective choice and social welfare. Holden Day, San Francisco
Sion M (1958): On general minimax theorems. Pacific Journal of Mathematics 8: 171–176
Tversky A (1969): Intransitivity of preferences. Psychological Review 76: 31–48
Uryas’ev SP (1990): Adaptive algorithms of stochastic optimization and game theory. Nauka, Moscow (in Russian)
Xiu N, Zhang J (2003): Some recent advances in projection-type methods for variational inequalities. Journal of Computational and Applied Mathematics 152: 559–585
Yao JC (1994): Multivalued variational inequalities with K-pseudomonotone operators. Journal of Optimization Theory and Applications 83: 391–403
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Konnov, I.V. (2007). Combined Relaxation Methods for Generalized Monotone Variational Inequalities. In: Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol 583. Springer, Berlin, Heidelberg . https://doi.org/10.1007/978-3-540-37007-9_1
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DOI: https://doi.org/10.1007/978-3-540-37007-9_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37006-2
Online ISBN: 978-3-540-37007-9
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