| Title:
|
Modified golden ratio algorithms for pseudomonotone equilibrium problems and variational inequalities (English) |
| Author:
|
Yin, Lulu |
| Author:
|
Liu, Hongwei |
| Author:
|
Yang, Jun |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
67 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
273-296 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We propose a modification of the golden ratio algorithm for solving pseudomonotone equilibrium problems with a Lipschitz-type condition in Hilbert spaces. A new non-monotone stepsize rule is used in the method. Without such an additional condition, the theorem of weak convergence is proved. Furthermore, with strongly pseudomonotone condition, the $R$-linear convergence rate of the method is established. The results obtained are applied to a variational inequality problem, and the convergence rate of the problem under the condition of error bound is considered. Finally, numerical experiments on several specific problems and comparison with other algorithms show the superiority of the algorithm. (English) |
| Keyword:
|
equilibrium problem |
| Keyword:
|
strongly pseudomonotone bifunctions |
| Keyword:
|
Lipschitz-type condition |
| Keyword:
|
variational inequality |
| Keyword:
|
error bound |
| MSC:
|
47J25 |
| MSC:
|
49J40 |
| MSC:
|
65K10 |
| MSC:
|
65K15 |
| MSC:
|
90C25 |
| MSC:
|
90C33 |
| MSC:
|
90C48 |
| MSC:
|
91B50 |
| idZBL:
|
Zbl 07547196 |
| idMR:
|
MR4409307 |
| DOI:
|
10.21136/AM.2021.0180-20 |
| . |
| Date available:
|
2022-04-14T13:35:01Z |
| Last updated:
|
2024-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150316 |
| . |
| Reference:
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[1] Bauschke, H. H., Combettes, P. L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces.CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011). Zbl 1218.47001, MR 2798533, 10.1007/978-1-4419-9467-7 |
| Reference:
|
[2] Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Nonlinear Programming Techniques for Equilibria.EURO Advanced Tutorials on Operational Research. Springer, Cham (2019). Zbl 06954058, MR 3838394, 10.1007/978-3-030-00205-3 |
| Reference:
|
[3] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems.Math. Stud. 63 (1994), 123-145. Zbl 0888.49007, MR 1292380 |
| Reference:
|
[4] Daniele, P., Giannessi, F., (eds.), A. Maugeri: Equilibrium Problems and Variational Models.Nonconvex Optimization and Its Applications 68. Kluwer, Dordrecht (2003). Zbl 1030.00031, MR 2042582, 10.1007/978-1-4613-0239-1 |
| Reference:
|
[5] Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. 1.Springer Series in Operations Research. Springer, New York (2003). Zbl 1062.90001, MR 1955648, 10.1007/b97543 |
| Reference:
|
[6] Fan, K.: A minimax inequality and applications.Inequalities. III Academic Press, New York (1972), 103-113. Zbl 0302.49019, MR 0341029 |
| Reference:
|
[7] am, S. D. Fl\accent23, Antipin, A. S.: Equilibrium programming using proximal-like algorithms.Math. Program. 78 (1997), 29-41. Zbl 0890.90150, MR 1454787, 10.1007/BF02614504 |
| Reference:
|
[8] Hieu, D. V.: Halpern subgradient extragradient method extended to equilibrium problems.Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 111 (2017), 823-840. Zbl 1378.65136, MR 3661152, 10.1007/s13398-016-0328-9 |
| Reference:
|
[9] Hieu, D. V.: Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems.Numer. Algorithms 77 (2018), 983-1001. Zbl 06860399, MR 3779075, 10.1007/s11075-017-0350-9 |
| Reference:
|
[10] Hieu, D. V.: The convergence rate of a golden ratio algorithm for equilibrium problems.Available at https://arxiv.org/abs/1810.03564 (2018), 11 pages. |
| Reference:
|
[11] Hieu, D. V.: New inertial algorithm for a class of equilibrium problems.Numer Algorithms 80 (2019), 1413-1436. Zbl 07042055, MR 3927239, 10.1007/s11075-018-0532-0 |
| Reference:
|
[12] Hieu, D. V., Cho, Y. J., Xiao, Y.-b.: Modified extragradient algorithms for solving equilibrium problems.Optimization 67 (2018), 2003-2029. Zbl 1416.90050, MR 3885897, 10.1080/02331934.2018.1505886 |
| Reference:
|
[13] Hieu, D. V., Cho, Y. J., Xiao, Y.-b., Kumam, P.: Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces.Optimization 69 (2020), 2279-2304. Zbl 1459.65096, MR 4156869, 10.1080/02331934.2019.1683554 |
| Reference:
|
[14] Hieu, D. V., Cho, Y. J., Xiao, Y.-b., Kumam, P.: Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces.Vietnam J. Math. 49 (2021), 1165-1183. Zbl 7425500, MR 4319545, 10.1007/s10013-020-00447-7 |
| Reference:
|
[15] Hieu, D. V., Strodiot, J. J., Muu, L. D.: Modified golden ratio algorithms for solving equilibrium problems.Available at https://arxiv.org/abs/1907.04013 (2019), 14 pages. |
| Reference:
|
[16] Hieu, D. V., Strodiot, J. J., Muu, L. D.: An explicit extragradient algorithm for solving variational inequalities.J. Optim. Theory Appl. 185 (2020), 476-503. Zbl 07198926, MR 4096353, 10.1007/s10957-020-01661-6 |
| Reference:
|
[17] Hieu, D. V., Thong, D. V.: New extragradient-like algorithms for strongly pseudomonotone variational inequalities.J. Glob. Optim. 70 (2018), 385-399. Zbl 1384.65041, MR 3761263, 10.1007/s10898-017-0564-3 |
| Reference:
|
[18] Kim, D. S., Vuong, P. T., Khanh, P. D.: Qualitative properties of strongly pseudomonotone variational inequalities.Optim. Lett. 10 (2016), 1669-1679. Zbl 1392.90115, MR 3556951, 10.1007/s11590-015-0960-x |
| Reference:
|
[19] Konnov, I. V.: Combined Relaxation Methods for Variational Inequalities.Lecture Notes in Economics and Mathematical Systems 495. Springer, Berlin (2001). Zbl 0982.49009, MR 1795730, 10.1007/978-3-642-56886-2 |
| Reference:
|
[20] Konnov, I. V.: Equilibrium Models and Variational Inequalities.Mathematics in Science and Engineering 210. Elsevier, Amsterdam (2007). Zbl 1140.91056, MR 2503647, 10.1016/s0076-5392(07)x8001-9 |
| Reference:
|
[21] Korpelevich, G. M.: An extragradient method for finding saddle points and other problems.Ehkon. Mat. Metody Russian 12 (1976), 747-756. Zbl 0342.90044, MR 0451121 |
| Reference:
|
[22] Malitsky, Y.: Golden ratio algorithms for variational inequalities.Math. Program. 184 (2020), 383-410. Zbl 07263698, MR 4037890, 10.1007/s10107-019-01416-w |
| Reference:
|
[23] Martinet, B.: Régularisation d'inéquations variationnelles par approximations successives.Rev. Franç. Inform. Rech. Opér. French 4 (1970), 154-158. Zbl 0215.21103, MR 298899, 10.1051/m2an/197004R301541 |
| Reference:
|
[24] Muu, L. D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria.Nonlinear Anal., Theory Methods Appl. 18 (1992), 1159-1166. Zbl 0773.90092, MR 1171603, 10.1016/0362-546X(92)90159-C |
| Reference:
|
[25] Muu, L. D., Quy, N. V.: On existence and solution methods for strongly pseudomonotone equilibrium problems.Vietnam J. Math. 43 (2015), 229-238. Zbl 1317.47058, MR 3349814, 10.1007/s10013-014-0115-x |
| Reference:
|
[26] Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables.Computer Science and Applied Mathematics. Academic Press, New York (1970). Zbl 0241.65046, MR 0273810, 10.1016/c2013-0-11263-9 |
| Reference:
|
[27] Rockafellar, R. T.: Monotone operators and the proximal point algorithm.SIAM J. Control Optim. 14 (1976), 877-898. Zbl 0358.90053, MR 0410483, 10.1137/0314056 |
| Reference:
|
[28] Tran, D. Q., Dung, M. L., Nguyen, V. H.: Extragradient algorithms extended to equilibrium problems.Optimization 57 (2008), 749-776. Zbl 1152.90564, MR 2473940, 10.1080/02331930601122876 |
| Reference:
|
[29] Vinh, N. T.: Golden ratio algorithms for solving equilibrium problems in Hilbert spaces.Available at https://arxiv.org/abs/1804.01829 (2018), 25 pages. |
| Reference:
|
[30] Yang, J., Liu, H.: A self-adaptive method for pseudomonotone equilibrium problems andvariational inequalities.Comput. Optim. Appl. 75 (2020), 423-440. Zbl 1432.49013, MR 4064596, 10.1007/s10589-019-00156-z |
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