zbMATH — the first resource for mathematics

Convergence analysis of projection methods for a new system of general nonconvex variational inequalities. (English) Zbl 06214014
Summary: In this article, we introduce and consider a new system of general nonconvex variational inequalities defined on uniformly prox-regular sets. We establish the equivalence between the new system of general nonconvex variational inequalities and the fixed point problems to analyze an explicit projection method for solving this system. We also consider the convergence of the projection method under some suitable conditions. Results presented in this article improve and extend the previously known results for the variational inequalities and related optimization problems.

47J20 Variational and other types of inequalities involving nonlinear operators (general)
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
Full Text: DOI
[1] Stampacchia, G, Formes bilinearies coercivities sur LES ensembles convexes, C R Acad Sci Paris, 258, 4413-4416, (1964) · Zbl 0124.06401
[2] Noor, MA, Some developments in general variational inequalities, Appl Math Comput, 152, 199-277, (2004) · Zbl 1134.49304
[3] Bounkhel, M; Tadj, L; Hamdi, A, Iterative schemes to solve nonconvex variational problems, J Inequal Pure Appl Math, 4, 1-14, (2003) · Zbl 1045.58014
[4] Noor, MA, On implicit methods for nonconvex variational inequalities, J Optim Theory Appl, 147, 411-417, (2010) · Zbl 1202.90253
[5] Pang, LP; Shen, J; Song, HS, A modified predictor-corrector algorithm for solving non-convex generalized variational inequalities, Comput Math Appl, 54, 319-325, (2007) · Zbl 1131.49010
[6] Verma, RU, Generalized system for relaxed cocoercive variational inequalities and projection methods, J Optim Theory Appl, 121, 203-210, (2004) · Zbl 1056.49017
[7] Noor, MA, Some iterative methods for nonconvex variational inequalities, Comput Math Model, 21, 97-108, (2010) · Zbl 1201.65114
[8] Clarke FH, Ledyaev YS, Wolenski PR: Nonsmooth Analysis and Control Theory. Springer, Berlin; 1998.
[9] Poliquin, RA; Rockafellar, RT; Thibault, L, Local differentiability of distance functions, Trans Am Math Soc, 352, 5231-5249, (2000) · Zbl 0960.49018
[10] Noor, MA, Projection methods for nonconvex variational inequalities, Optim Lett, 3, 411-418, (2009) · Zbl 1171.58307
[11] Noor, MA, Iterative methods for general nonconvex variational inequalities, Albanian J Math, 3, 117-127, (2009) · Zbl 1213.49017
[12] Huang, Z; Noor, MA, An explicit projection method for a system of nonlinear variational inequalities with different (\(γ\), \(r\))-cocoercive mappings, Appl Math Comput, 190, 356-361, (2007) · Zbl 1120.65080
[13] Verma, RU, General convergence analysis for two-step projection methods and applications to variational problems, Appl Math Lett, 18, 1286-1292, (2005) · Zbl 1099.47054
[14] Noor, MA; Noor, KI, Projection algorithms for solving system of general variational inequalities, Nonlinear Anal, 70, 2700-2706, (2009) · Zbl 1156.49010
[15] Wen, DJ, Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators, Nonlinear Anal, 73, 2292-2297, (2010) · Zbl 1229.47104
[16] Ceng, LC; Teboulle, M; Yao, JC, Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems, J Optim Theory Appl, 146, 19-31, (2010) · Zbl 1222.47091
[17] Wen, DJ, Strong convergence theorems for equilibrium problems and \(k\)-strict pseudo-contractions in Hilbert spaces, No. 2011, (2011)
[18] Xu, HK, Iterative algorithms for nonlinear operators, J Lond Math Soc, 2, 240-256, (2002) · Zbl 1013.47032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.