Contraction of the proximal mapping and applications to the equilibrium problem. (English) Zbl 1358.90142
Summary: In this paper, we introduce a new definition of Lipschitz-type continuity of a bifunction. Using this definition, we prove the contraction of the proximal mapping and apply it to the equilibrium problem over the fixed-point set of a nonexpansive mapping. We present a new algorithm for this problem. Under classical conditions, the convergence of the algorithm is proved. Finally, we present some numerical results for the proposed algorithm.
MSC:
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
| 65K10 | Numerical optimization and variational techniques |
| 90C25 | Convex programming |
Keywords:
proximal mapping; equilibrium problem; variational inequality problem; fixed-point problem; contractive mapping; nonexpansive mappingReferences:
| [1] | Blum E, Math Student 63 pp 123– (1994) |
| [2] | DOI: 10.1073/pnas.36.1.48 · Zbl 0036.01104 · doi:10.1073/pnas.36.1.48 |
| [3] | DOI: 10.2307/1969529 · Zbl 0045.08202 · doi:10.2307/1969529 |
| [4] | Fan K, Inequality III (1972) |
| [5] | DOI: 10.1007/BF02192244 · Zbl 0903.49006 · doi:10.1007/BF02192244 |
| [6] | DOI: 10.1007/978-1-4613-0299-5 · doi:10.1007/978-1-4613-0299-5 |
| [7] | DOI: 10.1007/b101428 · doi:10.1007/b101428 |
| [8] | DOI: 10.1016/j.ejor.2012.11.037 · Zbl 1292.90315 · doi:10.1016/j.ejor.2012.11.037 |
| [9] | Combettes LP, J Nonlinear Convex Anal 6 pp 117– (2005) |
| [10] | DOI: 10.1016/j.jmaa.2007.02.044 · Zbl 1127.47053 · doi:10.1016/j.jmaa.2007.02.044 |
| [11] | DOI: 10.1016/j.na.2008.02.042 · Zbl 1142.47350 · doi:10.1016/j.na.2008.02.042 |
| [12] | Kinderlehrer D, An Introduction to variational inequalities and their applications (1980) · Zbl 0457.35001 |
| [13] | DOI: 10.1016/S1570-579X(01)80028-8 · doi:10.1016/S1570-579X(01)80028-8 |
| [14] | DOI: 10.1007/978-1-4612-5020-3 · doi:10.1007/978-1-4612-5020-3 |
| [15] | DOI: 10.24033/bsmf.1625 · Zbl 0136.12101 · doi:10.24033/bsmf.1625 |
| [16] | Moreau JJ, C. R. Acad Sci Paris 256 pp 1069– (1963) |
| [17] | DOI: 10.1017/CBO9780511526152 · doi:10.1017/CBO9780511526152 |
| [18] | DOI: 10.1007/978-1-4613-0239-1 · doi:10.1007/978-1-4613-0239-1 |
| [19] | Rodriguez V, Proceedings of IEEE Wireless Communication Network Conference; 2003 Mar; NewOrleans, LA pp 717– (2003) |
| [20] | DOI: 10.1109/26.983324 · doi:10.1109/26.983324 |
| [21] | DOI: 10.1109/26.935162 · Zbl 1013.94502 · doi:10.1109/26.935162 |
| [22] | DOI: 10.1007/s10852-008-9099-4 · Zbl 1170.93020 · doi:10.1007/s10852-008-9099-4 |
| [23] | DOI: 10.1007/s10107-010-0427-x · Zbl 1274.90428 · doi:10.1007/s10107-010-0427-x |
| [24] | DOI: 10.1080/02331930701762829 · Zbl 1158.47312 · doi:10.1080/02331930701762829 |
| [25] | Rockafellar RT, Variational analysis, Fundamental principles of mathematical sciences. Vol. 317 (1998) |
| [26] | DOI: 10.1023/A:1026050425030 · Zbl 1061.90112 · doi:10.1023/A:1026050425030 |
| [27] | DOI: 10.1080/02331930601122876 · Zbl 1152.90564 · doi:10.1080/02331930601122876 |
| [28] | DOI: 10.1007/s10898-011-9693-2 · Zbl 1258.90088 · doi:10.1007/s10898-011-9693-2 |
| [29] | Agarwal RP, Fixed point theory for Lipschitzian-type mappings with applications (2009) |
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