Abstract
Potential equilibrium problems are considered. The notions of bilinear differential and bi-convexity are introduced. The concept of generalized potentiality is offered. The convergence of gradient prediction-type methods for solving of generalized potential equilibrium problems is justified. Estimates of convergence rate are derived.
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Antipin, A.S. (1977), A method of searching of saddle point for augmented Lagrange function, Ekon. Math. Metody 3: 560–565.
Antipin, A.C. (1992), Controlled Proximal Differential Systems for Saddle Problems, Differentsial, nye Uravneniya 11: 1846–1861. English transl.: (1992),Differential Equations 11: 1498–1510.
Antipin, A.C. (1992), Inverse optimization problem: statement and approaches to solving it, Inverse Mathematical Programming Problems. Computing Center of Russian Academy of Sciences. 6-56.: English transl.: (1996), Computational Mathematics and Modelling. Plenum, New York, 3: 263–287.
Antipin, A.S. (1994), Feedback-Controlled Saddle Gradient Processes, Avtomatika i telemechanika 3: 12-23. English transl: (1994), Automation and Remote Control 3: 311–320.
Antipin, A.C. (1995), Differential gradient prediction-type methods for computing fixed points of extreme maps, Differentsial'nye Uravneniya 11: 1786–1795. English transl.: (1995), Differential Equations 11: 1754–11763.
Antipin, A.C. (1995), On convergence rate estimation for gradient projection method, Izvest. VUZov.Matematika 6: 16–24. English transl.: (1995), RussianMathematics (Iz.VUZ) 6: 14–22.
Antipin, A.S. (1995), The convergence of proximal methods to fixed points of extreme mappings and estimates of their rate of convergence, Zhurnal vychisl. mat. i mat. fiz 5: 688–704. English transl.: (1995), Comp. Maths. Math. Phys 5: 539–551.
Antipin, A.S. (1998), Splitting of a gradient approach for solving extreme inclusions, Zhurnal vychisl. mat. i mat. fiz 7: 1118–1132. English transl.: (1998), Comp. Maths. Math. Phys. 7: 1069–1082.
Aubin, J.-P. and Frankowska, H.(1990), Set Valued Analysis, Birkhauser.
Belen'ky, V.Z. and Volkonsky, V.A. (1974), Iterative Methods in theory of game and programming, Nauka, Moscow (in Russian).
Bianchi, M. and Schaible, S. (1996), Generalized monotone bifunctions and equilibrium problems. J. Optimiz. Theory and Appl 1: 31–43.
Blum, E. and Oettli, W. (1993), From Optimization and Variational Inequalities to Equilibrium Problems, The Mathematics Student 1–4: 1–23.
Elster K.-H. (1977), Einfuhrung in die nichtlineare optimirung BSB B.G. Teubner Verlagsgesellschat. Leipzig.
Fan, Ky. (1972), A minimax inequality and applications, Inequalities III. Academic Press, New York, 103–113.
Flam, S.D. (1993), Paths to constrained Nash equilibria. Applied Mathematics and Optimization 27: 275–289.
Golshtein, E.G. and Tretyakov, N.V. (1996), Modified Lagrangians and Monotone Maps in Optimization John Wiley, New York.
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 48: 161–220.
Kaplan, A., and Tichatschke, R.(1997), Multi-Step Proximal Method for Variational Inequalities with Monotone Operators. Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems 452. Springer, Berlin.
Korpelevich, G.M. (1976), Extragradient method for finding the saddle points and for other problems. Ekon. Math. Metody 4: 747–756.
Mangasarian, O.L. (1994), Nonlinear Programming SIAM. Philadelphia, PA.
Migdalas, A. and Pardalos, P.M. (1996), Editorial: Hierarchical and Bilevel Programming. J. of Global Optimization 8: 209–215.
Monderer, D. and Shapley, L.S. (1996), Potential games. Games and economic behavior 14: 124–143.
Nash J.F. Jr. (1950), Equilibrium points in n-person games. Proc. Nat.Acad.Sci. USA 36: 48–49.
Nesterov, Yu.E. (1989), Effective Methods in nonlinear programming Moscow, Radio and Svyaz.
Ortega, J.M. and Rheinbold, W.C. (1970), Iterative solution of nonlinear equations in several variables Academic Press, New York, London.
Polyak, B.Th. (1973), Introduction to optimization, Moscow.
Rockafellar, R.T. (1970), Convex Analysis. Princeton University Press.
Rosen, J.B. (1965), Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 3: 520–534.
Vasil'ev, F.P. (1988), Numerical Methods of solving for extreme problems Nauka. Moscow (in Russian).
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Antipin, A. Gradient Approach of Computing Fixed Points of Equilibrium Problems. Journal of Global Optimization 24, 285–309 (2002). https://doi.org/10.1023/A:1020321209606
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DOI: https://doi.org/10.1023/A:1020321209606