A two-stage proximal algorithm for equilibrium problems in Hadamard spaces

Vedel, YI
1Semenov, VV
2Chabak, LM
1V.М. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv
2Hetman Petro Konashevich-Sahaydachniy Kyiv State Maritime Academy
Dopov. Nac. akad. nauk Ukr. 2020, 2:7-14
https://doi.org/10.15407/dopovidi2020.02.007
Section: Information Science and Cybernetics
Language: Russian
Abstract: 

We consider the equilibrium problem in Hadamard spaces, which extends and unifies several problems in optimization, variational inequalities, fixed-point theory, and many other parts in nonlinear analysis. First, we give the necessary facts about Hadamard metric spaces and consider the statements of equilibrium problems associated with pseudo-monotone bifunctions with suitable conditions on the bifunctions in Hadamard spaces. Then, to approximate an equilibrium point, we consider the two-stage proximal algorithm for pseudo-monotone bifunctions. This algorithm is an analog of the previously studied two-stage algorithm for equilibrium problems in a Hilbert space. For Lipschitz-type pseudo-monotone bifunctions, a theorem on the weak convergence of sequences generated by the algorithm is proved.

Keywords: convergence., equilibrium problem, Hadamard space, pseudo-monotonicity, two-stage algorithm
References: 

1. Antipin, A. S. (1997). Equilibrium programming: Proximal methods. Comput. Math. Math. Phys., 37, pp. 1285-1296. Doi: https://doi.org/10.1134/S0965542507120044
2. Mastroeni, G. (2003). On auxiliary principle for equilibrium problems. In: Daniele, P. et al. (eds.) Equilibrium Problems and Variational Models. Dordrecht: Kluwer Acad. Publ., pp. 289-298. Doi: https://doi.org/10.1007/978-1-4613-0239-1
3. Combettes, P. L. & Hirstoaga, S. A. (2005). Equilibrium Programming in Hilbert Spaces. J. Nonlinear Convex Anal., 6, pp. 117-136.
4. Korpelevich, G. M. (1976). An extragradient method for finding saddle points and for other problems. Matecon, 12, No. 4, pp. 747-756.
5. Quoc, T. D., Muu, L. D. & Hien, N. V. (2008). Extragradient algorithms extended to equilibrium problems. Optimization, 57, pp. 749-776. Doi: https://doi.org/10.1080/02331930601122876
6. Popov, L. D. (1980). A modification of the Arrow-Hurwicz method for search of saddle points. Mathematical notes of the Academy of Sciences of the USSR, 28, Iss. 5, pp. 845-848. Doi: https://doi.org/10.1007/BF01141092
7. Lyashko, S. I. & Semenov, V. V. (2016). A New Two-Step Proximal Algorithm of Solving the Problem of Equilibrium Programming. In: Goldengorin, B. (ed.) Optimization and Its Applications in Control and Data Sciences. Springer Optimization and Its Applications, vol. 115. Springer, Cham, pp. 315-325. Doi: https://doi.org/10.1007/978-3-319-42056-1_10
8. Bacak, M. (2014). Convex Analysis and Optimization in Hadamard Spaces. Berlin-Boston: De Gruyter, viii+185 p. Doi: https://doi.org/10.1515/9783110361629
9. Colao, V., Lopez, G., Marino, G. & Martin-Marquez, V. (2012). Equilibrium problems in Hadamard manifolds. Journal of Mathematical Analysis and Applications, 388, pp. 61-77. Doi: https://doi.org/10.1016/j.jmaa.2011.11.001
10. Khatibzadeh, H. & Mohebbi, V. (2019). Monotone and pseudo-monotone equilibrium problems in Hada mard spaces. J. the Australian Mathematical Society. pp. 1-23. Doi: https://doi.org/10.1017/S1446788719000041
11. Khatibzadeh, H. & Mohebbi, V. (2019). Approximating solutions of equilibrium problems in Hadamard spaces. Miskolc Mathematical Notes, 20, No. 1, pp. 281-297. Doi: https://doi.org/10.18514/MMN.2019.2361
12. Chabak, L., Semenov, V., Vedel, Y. (2019). A New Non-Euclidean Proximal Method for Equilibrium Problems. In: Chertov O., Mylovanov T., Kondratenko Y., Kacprzyk J., Kreinovich V. & Stefanuk V. (eds.) Recent Developments in Data Science and Intelligent Analysis of Information. ICDSIAI 2018. Advances in Intelligent Systems and Computing, vol. 836. Springer, Cham, pp. 50-58. Doi: https://doi.org/10.1007/978-3-319-97885-7_6
13. Semenov, V. V. (2017). A Version of the Mirror descent Method to Solve Variational Inequalities. Cybernetics and Systems Analysis, 53. Iss. 2, pp. 234-243. Doi: https://doi.org/10.1007/s10559-017-9923-9