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Extragradient methods for solving nonconvex variational inequalities. (English) Zbl 1211.65082

This paper is devoted to the study of a new class of variational inequalities (the nonconvex variational inequalities), and for a new class of nonconvex sets (uniformly prox-regular set K r ). This class of uniformly prox-regular sets have played an important part in many nonconvex applications (optimization, dynamic systems and differential inclusions).

For a nonlinear operator T the authors establish the equivalence between the variational inequality: <Tu,v-u>0, [u,v]K r (the nonconvex variational inequality, NVI) and the fixed point problem using the projection operator technique. Here uK r is a solution of the nonconvex variational inequality if and only if uK r satisfies the relation: u=P K r [u-ρTu], where P K r is the projection of H (a real Hilbert space) onto the uniformly prox-regular set K r , which implies that NVI is equivalent to the fixed point problem. This equivalent formulation is used to suggest and analyze the implicit iterative method for solving NVI. If the operator T is pseudomonotone, and uK r is a solution of NVI and u n+1 is the approximate solution obtained from Algorithm 2 (For a given u H find the approximate solution u n+1 by using the iterative schemes: u n+1 =P K r [u n -ρTu n+1 ],n=0,1,···) one then has: u-u n+1 2 u-u n 2 -u n+1 -u n 2 , ρ>0, and lim n u n =u, if H is a finite dimensional space.

The authors use the idea of Noor to prove that the convergence of the extragradient method requires only pseudo-monotonicity, which is a weaker condition than monotonicity. Thus proposed result represents an improvement and refinement of the known results.

MSC:
65K15Numerical methods for variational inequalities and related problems
49J40Variational methods including variational inequalities
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
References:
[1]Stampacchia, G.: Formes bilineaires coercitives sur LES ensembles convexes, C. R. Acad. sci. Paris 258, 4413-4416 (1964) · Zbl 0124.06401
[2]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 · doi:emis:journals/JIPAM/v4n1/index.html
[3]Clarke, F. H.; Ledyaev, Y. S.; Wolenski, P. R.: Nonsmooth analysis and control theory, (1998)
[4]Kinderlehrer, D.; Stampacchia, G.: An introduction to variational inequalities and their applications, (2000)
[5]Korpelevich, G. M.: An extragradient method for finding saddle points and for other problems, Matecon 12, 747-756 (1976)
[6]Lions, P. L.; Mercier, B.: Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. anal. 16, 964-979 (1979) · Zbl 0426.65050 · doi:10.1137/0716071
[7]M. Aslam Noor, On variational inequalities, Ph.D. Thesis, Brunel University, London, UK, 1975.
[8]Noor, M. Aslam: General variational inequalities, Appl. math. Lett. 1, 119-121 (1988) · Zbl 0655.49005 · doi:10.1016/0893-9659(88)90054-7
[9]Noor, M. Aslam: Some recent advances in variational inequalities, part II, other concepts, New Zealand J. Math. 26, 229-255 (1997) · Zbl 0889.49006
[10]Noor, M. Aslam: New approximation schemes for general variational inequalities, J. math. Anal. appl. 251, 217-229 (2000) · Zbl 0964.49007 · doi:10.1006/jmaa.2000.7042
[11]Noor, M. Aslam: Some developments in general variational inequalities, Appl. math. Comput. 152, 199-277 (2004) · Zbl 1134.49304 · doi:10.1016/S0096-3003(03)00558-7
[12]Noor, M. Aslam: Iterative schemes for nonconvex variational inequalities, J. optim. Theory appl. 121, 385-395 (2004) · Zbl 1062.49009 · doi:10.1023/B:JOTA.0000037410.46182.e2
[13]Noor, M. Aslam: Fundamentals of mixed quasi variational inequalities, Int. J. Pure appl. Math. 15, 137-258 (2004) · Zbl 1059.49018
[14]Noor, M. Aslam: Projection methods for nonconvex variational inequalities, Optim. lett. 3, 411-418 (2009) · Zbl 1171.58307 · doi:10.1007/s11590-009-0121-1
[15]Noor, M. Aslam: Implicit iterative method for nonconvex variational inequalities, J. optim. Theory appl. 143, 619-624 (2009) · Zbl 1187.90297 · doi:10.1007/s10957-009-9567-7
[16]Noor, M. Aslam: An extragradient algorithm for solving the general nonconvex variational inequalities, Appl. math. Lett. 23, 917-921 (2010) · Zbl 1193.49008 · doi:10.1016/j.aml.2010.04.011
[17]Noor, M. Aslam: On an implicit method for nonconvex variational inequalities, J. optim. Theory appl. 147, 411-417 (2010) · Zbl 1202.90253 · doi:10.1007/s10957-010-9717-y
[18]Noor, M. Aslam: New implicit methods for general nonconvex variational inequalities, Bull. math. Anal. appl. 3 (2010)
[19]Noor, M. Aslam: Iterative methods for general nonconvex variational inequalities, Albanian J. Math. 3, 117-127 (2009) · Zbl 1213.49017 · doi:http://x.kerkoje.com/index.php/ajm/article/viewArticle/134
[20]Noor, M. Aslam: Some iterative methods for general nonconvex variational inequalities, Comput. math. Model. 21, 97-108 (2010) · Zbl 1201.65114 · doi:10.1007/s10598-010-9057-7
[21]Noor, M. Aslam; Noor, K. Inayat; Rassias, Th.M.: Some aspects of variational inequalities, J. comput. Appl. math. 47, 285-312 (1993) · Zbl 0788.65074 · doi:10.1016/0377-0427(93)90058-J
[22]Poliquin, R. A.; Rockafellar, R. T.; Thibault, L.: Local differentiability of distance functions, Trans. amer. Math. soc. 352, 5231-5249 (2000) · Zbl 0960.49018 · doi:10.1090/S0002-9947-00-02550-2