# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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: $<Tu, v-u> \geq 0$, $[u,v] \in K_r$ (the nonconvex variational inequality, NVI) and the fixed point problem using the projection operator technique. Here $u \in K_r$ is a solution of the nonconvex variational inequality if and only if $u \in K_r$ satisfies the relation: $u=P_{K_r} [u - \rho T u]$, 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 $u \in K_r$ is a solution of NVI and $u_{n+1}$ is the approximate solution obtained from Algorithm 2 (For a given $u_\diamond \in H$ find the approximate solution $u_{n+1}$ by using the iterative schemes: $u_{n+1} = P_{K_r} [u_n - \rho T u_{n+1}], n = 0,1, ...)$ one then has: $\|u-u_{n+1}\|^2 \leq \|u - u_n\|^2 - \|u_{n+1} - u_n \|^2$, $\rho >0$, and $\lim_{n \rightarrow \infty} u_n = u$, 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:
 65K15 Numerical methods for variational inequalities and related problems 49J40 Variational methods including variational inequalities 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions)
Full Text:
##### 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 · emis:journals/JIPAM/v4n1/index.html [3] Clarke, F. H.; Ledyaev, Y. S.; Wolenski, P. R.: Nonsmooth analysis and control theory, (1998) · Zbl 1047.49500 [4] Kinderlehrer, D.; Stampacchia, G.: An introduction to variational inequalities and their applications, (2000) · Zbl 0988.49003 [5] Korpelevich, G. M.: An extragradient method for finding saddle points and for other problems, Matecon 12, 747-756 (1976) · Zbl 0342.90044 [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) · Zbl 1312.49009 [19] Noor, M. Aslam: Iterative methods for general nonconvex variational inequalities, Albanian J. Math. 3, 117-127 (2009) · Zbl 1213.49017 · 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