Top

Published in:

2016 | OriginalPaper | Chapter

A New Two-Step Proximal Algorithm of Solving the Problem of Equilibrium Programming

Authors : Sergey I. Lyashko, Vladimir V. Semenov

Published in: Optimization and Its Applications in Control and Data Sciences

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We propose a new iterative two-step proximal algorithm for solving the problem of equilibrium programming in a Hilbert space. This method is a result of extension of L.D. Popov’s modification of Arrow-Hurwicz scheme for approximation of saddle points of convex-concave functions. The convergence of the algorithm is proved under the assumption that the solution exists and the bifunction is pseudo-monotone and Lipschitz-type.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Modeling of Stationary Periodic Time Series by ARMA Representations

next chapter Nonparametric Ellipsoidal Approximation of Compact Sets of Random Points

Anh, P.N.: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J. Optim. Theory Appl. 154, 303–320 (2012)MathSciNetCrossRefMATH

Antipin, A.S.: Equilibrium programming: gradient methods. Autom. Remote Control 58 (8), Part 2, 1337–1347 (1997)

Antipin, A.S.: Equilibrium programming: proximal methods. Comput. Math. Math. Phys. 37, 1285–1296 (1997)MathSciNetMATH

Antipin, A.: Equilibrium programming problems: prox-regularization and prox-methods. In: Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 452, pp. 1–18. Springer, Heidelberg (1997)

Antipin, A.S.: Extraproximal approach to calculating equilibriums in pure exchange models. Comput. Math. Math. Phys. 46, 1687–1998 (2006)MathSciNetCrossRef

Antipin, A.S.: Multicriteria equilibrium programming: extraproximal methods. Comput. Math. Math. Phys. 47, 1912–1927 (2007)MathSciNetCrossRef

Antipin, A.S., Vasil’ev, F.P., Shpirko, S.V.: A regularized extragradient method for solving equilibrium programming problems. Comput. Math. Math. Phys. 43 (10), 1394–1401 (2003)MathSciNetMATH

Antipin, A.S., Artem’eva, L.A., Vasil’ev, F.P.: Multicriteria equilibrium programming: extragradient method. Comput. Math. Math. Phys. 50 (2), 224–230 (2010)MathSciNetCrossRefMATH

Antipin, A.S., Jacimovic, M., Mijailovic, N.: A second-order continuous method for solving quasi-variational inequalities. Comput. Math. Math. Phys. 51 (11), 1856–1863 (2011)MathSciNetCrossRefMATH

10.

Antipin, A.S., Artem’eva, L.A., Vasil’ev, F.P.: Extraproximal method for solving two-person saddle-point games. Comput. Math. Math. Phys. 51 (9), 1472–1482 (2011)MathSciNetCrossRefMATH

11.

Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefMATH

12.

Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)MathSciNetMATH

13.

Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148 (2), 318–335 (2011)MathSciNetCrossRefMATH

14.

Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6 (1), 117–136 (2005)MathSciNetMATH

15.

Denisov, S.V., Semenov, V.V., Chabak, L.M.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern. Syst. Anal. 51, 757–765 (2015)MathSciNetCrossRefMATH

16.

Flam, S.D., Antipin, A.S.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997)MathSciNetCrossRefMATH

17.

Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer Academic, New York (2004)CrossRefMATH

18.

Iusem, A.N., Sosa, W.: On the proximal point method for equilibrium problems in Hilbert spaces. Optimization 59, 1259–1274 (2010)MathSciNetCrossRefMATH

19.

Khobotov, E.N.: Modification of the extra-gradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27 (5), 120–127 (1987)MathSciNetCrossRefMATH

20.

Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003)MathSciNetCrossRefMATH

21.

Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12, 747–756 (1976) (In Russian)MathSciNetMATH

22.

Lyashko, S.I., Semenov, V.V., Voitova, T.A.: Low-cost modification of Korpelevich’s methods for monotone equilibrium problems. Cybern. Syst. Anal. 47, 631–639 (2011)MathSciNetCrossRefMATH

23.

Malitsky, Yu.V., Semenov, V.V.: An extragradient algorithm for monotone variational inequalities. Cybern. Syst. Anal. 50, 271–277 (2014)MathSciNetCrossRefMATH

24.

Malitsky, Yu.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193–202 (2015)MathSciNetCrossRefMATH

25.

Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., et al. (eds.) Equilibrium Problems and Variational Models, pp. 289–298. Kluwer Academic, Dordrecht (2003)CrossRef

26.

Moudafi, A.: Proximal point methods extended to equilibrium problems. J. Nat. Geom. 15, 91–100 (1999)MathSciNetMATH

27.

Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006)MathSciNetCrossRefMATH

28.

Nurminski, E.A.: The use of additional diminishing disturbances in Fejer models of iterative algorithms. Comput. Math. Math. Phys. 48, 2154–2161 (2008)MathSciNetCrossRef

29.

Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)MathSciNetCrossRefMATH

30.

Popov, L.D.: A modification of the Arrow-Hurwicz method for search of saddle points. Math. Notes Acad. Sci. USSR 28 (5), 845–848 (1980)CrossRefMATH

31.

Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)MathSciNetCrossRef

32.

Semenov, V.V.: On the parallel proximal decomposition method for solving the problems of convex optimization. J. Autom. Inf. Sci. 42 (4), 13–18 (2010)CrossRef

33.

Semenov, V.V.: A strongly convergent splitting method for systems of operator inclusions with monotone operators. J. Autom. Inf. Sci. 46 (5), 45–56 (2014)CrossRef

34.

Semenov, V.V.: Hybrid splitting methods for the system of operator inclusions with monotone operators. Cybern. Syst. Anal. 50, 741–749 (2014)MathSciNetCrossRefMATH

35.

Semenov, V.V.: Strongly convergent algorithms for variational inequality problem over the set of solutions the equilibrium problems. In: Zgurovsky, M.Z., Sadovnichiy, V.A. (eds.) Continuous and Distributed Systems, pp. 131–146. Springer, Heidelberg (2014)CrossRef

36.

Stukalov, A.S.: An extraproximal method for solving equilibrium programming problems in a Hilbert space. Comput. Math. Math. Phys. 46, 743–761 (2006)MathSciNetCrossRef

37.

Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)MathSciNetCrossRefMATH

38.

Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38 (2), 431–446 (2000)MathSciNetCrossRefMATH

39.

Verlan, D.A., Semenov, V.V., Chabak, L.M.: A strongly convergent modified extragradient method for variational inequalities with non-Lipschitz operators. J. Autom. Inf. Sci. 47 (7), 31–46 (2015)CrossRef

40.

Vuong, P.T., Strodiot, J.J, Nguyen, V.H.: Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J. Optim. Theory Appl. 155, 605–627 (2012)MathSciNetCrossRefMATH

Title: A New Two-Step Proximal Algorithm of Solving the Problem of Equilibrium Programming
Authors: Sergey I. Lyashko
Vladimir V. Semenov
Publisher: Springer International Publishing
Book: Optimization and Its Applications in Control and Data Sciences
Print ISBN: 978-3-319-42054-7

Electronic ISBN: 978-3-319-42056-1

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-3-319-42056-1_10

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"